1. Budgeting and Money

What Is a Budget?

A budget is a plan for how money will be received, spent, and saved over a particular period of time.

A budget might cover:

• one week
• one month
• one year
• a particular event
• a project
• a trip

The basic idea is:

Income − Expenses = Money Remaining

The money remaining could be saved, invested, kept for future expenses, or used for additional spending.

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Why Do People Use Budgets?

Money is limited, but people usually have many possible ways to use it.

A budget helps organize these choices.

For example, someone might need money for:

• housing
• food
• transportation
• utilities
• education
• entertainment
• savings
• unexpected expenses

A budget allows a person to compare available income with planned expenses before making financial decisions.

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Income

Income is money received.

Examples can include:

• wages or salary
• allowance
• business income
• scholarships
• interest
• investment income
• gifts
• freelance work

Suppose someone earns:

$600 per month

from part-time work and receives:

$100 per month

from another source.

Total income:

$600 + $100 = $700

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Expenses

An expense is money that is spent.

Suppose monthly expenses are:

Total expenses:

80 + 180 + 40 + 70 + 50 + 30

= $450

If income is $700:

$700 − $450 = $250

There is:

$250 remaining

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Income, Expenses, and Balance

A useful budget equation is:

Balance = Income − Expenses

If:

Income > Expenses

there is a surplus.

If:

Income = Expenses

the budget is balanced, but there is no remaining amount.

If:

Expenses > Income

there is a deficit.

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Example: Budget Surplus

Monthly income:

$1,200

Monthly expenses:

$950

Calculate:

$1,200 − $950 = $250

The budget has a:

$250 surplus

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Example: Budget Deficit

Monthly income:

$900

Monthly expenses:

$1,050

Calculate:

$900 − $1,050 = −$150

The budget has a:

$150 deficit

This means planned spending is $150 greater than income.

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Creating a Simple Budget

A useful budgeting process is:

Step 1: Determine total income.

Step 2: List expected expenses.

Step 3: Calculate total expenses.

Step 4: Subtract expenses from income.

Step 5: Decide how much money should be saved.

Step 6: Adjust spending if necessary.

Step 7: Compare actual spending with the original plan.

A budget is useful only if it is realistic and updated when circumstances change.

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Fixed and Variable Expenses

Expenses can behave differently.

A fixed expense generally stays the same over a particular period.

Examples might include:

• rent
• subscriptions
• loan payments
• some insurance payments

A variable expense can change.

Examples include:

• groceries
• transportation
• entertainment
• electricity
• clothing

Knowing which expenses can change makes it easier to adjust a budget.

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Needs and Wants

Another useful way to examine spending is to distinguish between needs and wants.

A need is something considered necessary for basic living or important responsibilities.

Examples may include:

• basic food
• housing
• necessary transportation
• essential clothing
• required school materials

A want is something desirable but not essential.

Examples might include:

• entertainment subscriptions
• restaurant meals
• upgraded electronics
• luxury clothing
• optional entertainment

The distinction is not always absolute. A particular expense may be a need for one person but a want for another depending on circumstances.

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Calculating Savings

Suppose monthly income is:

$1,500

and expenses are:

$1,180

Money remaining:

$1,500 − $1,180 = $320

If all of the remaining money is saved:

Savings = $320

If only $200 is saved:

$320 − $200 = $120

remains available for other purposes.

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Savings Rate

Savings can also be expressed as a percentage of income.

The basic calculation is:

Savings rate = Savings / Income × 100%

Suppose:

Income = $2,000

Savings = $300

Calculate:

300 / 2000 × 100% = 15%

Therefore:

Savings rate = 15%

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Finding Savings from a Percentage

Suppose someone decides to save:

20% of $1,800

Convert:

20% = 0.20

Calculate:

0.20 × $1,800 = $360

Therefore:

Savings = $360

Money available after setting aside savings:

$1,800 − $360 = $1,440

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Using Percentages to Build a Budget

Percentages can help divide income into categories.

Suppose monthly income is:

$2,000

A hypothetical budget allocates:

Housing = 35%

Food = 15%

Transportation = 10%

Savings = 20%

Other spending = 20%

The amounts would be:

Housing:

0.35 × 2000 = $700

Food:

0.15 × 2000 = $300

Transportation:

0.10 × 2000 = $200

Savings:

0.20 × 2000 = $400

Other:

0.20 × 2000 = $400

Total:

$2,000

These percentages are illustrative rather than a rule for how everyone should budget.

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Visualizing a Budget

A pie chart is useful for showing how a total amount is divided among categories.

2. Discounts and Sales

What Is a Discount?

A discount is a reduction in the original price of a product or service.

Stores often advertise discounts using percentages:

10% off

25% off

40% off

50% off

The percentage tells us what fraction of the original price will be removed.

For example:

25% off

means that 25% of the original price is subtracted.

The customer therefore pays:

75% of the original price

because:

100% − 25% = 75%

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Original Price, Discount, and Sale Price

There are three important quantities in a discount problem.

Original price: Price before the discount.

Discount amount: Amount removed from the original price.

Sale price: Price after the discount has been removed.

The basic relationship is:

Sale Price = Original Price − Discount Amount

For example:

Original price = $80

Discount = $20

Sale price:

$80 − $20 = $60

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Calculating a Percentage Discount

To calculate a discount:

Discount Amount = Original Price × Discount Rate

Remember to convert the percentage to a decimal.

For example:

20% = 0.20

Suppose an item costs:

$50

with:

20% off

Calculate:

$50 × 0.20 = $10

Discount:

$10

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Finding the Final Price

Once we know the discount:

Sale Price = Original Price − Discount

For the previous example:

Original price:

$50

Discount:

$10

Therefore:

$50 − $10 = $40

Final price:

$40

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Example: 30% Discount

A jacket originally costs:

$120

It is discounted by:

30%

Step 1: Convert the percentage.

30% = 0.30

Step 2: Calculate the discount.

$120 × 0.30 = $36

Step 3: Subtract the discount.

$120 − $36 = $84

Therefore:

Discount = $36

Final price = $84

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Using Fractions for Familiar Discounts

Some percentages are easy to calculate as fractions.

50% = 1/2

25% = 1/4

75% = 3/4

20% = 1/5

10% = 1/10

For example:

25% off $80

Since:

25% = 1/4

calculate:

$80 ÷ 4 = $20

Discount:

$20

Sale price:

$80 − $20 = $60

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Using Benchmark Percentages

Some discounts can be calculated mentally by combining familiar percentages.

For example:

15% = 10% + 5%

Find 15% of $60.

10%:

$60 ÷ 10 = $6

5%:

$6 ÷ 2 = $3

Therefore:

15% of $60 = $9

Sale price:

$60 − $9 = $51

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Another Mental Strategy

Find:

35% of $200

We can use:

30% + 5%

30%:

$60

5%:

$10

Therefore:

35% = $70

If this is a discount:

$200 − $70 = $130

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Using a Sale-Price Multiplier

There is a faster way to calculate the final price.

If an item is:

20% off

then the customer pays:

100% − 20% = 80%

Convert:

80% = 0.80

Therefore:

Sale Price = Original Price × 0.80

Example:

$150 × 0.80 = $120

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Discount Multipliers

Some useful examples are:

For a 35% discount:

100% − 35% = 65%

Therefore, multiply the original price by:

0.65

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Example Using a Multiplier

Original price:

$240

Discount:

35%

Percentage paid:

65%

Calculate:

$240 × 0.65 = $156

Therefore:

Final price = $156

Savings:

$240 − $156 = $84

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Calculating Savings

The savings from a discount are normally equal to the discount amount.

Suppose:

Original price = $90

Discount = 20%

Calculate:

$90 × 0.20 = $18

Therefore:

Savings = $18

Final price:

$90 − $18 = $72

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Discount Amount vs Final Price

These are easy to confuse.

Suppose:

Original price = $200

Discount = 30%

Discount amount:

$200 × 0.30 = $60

But the final price is:

$200 − $60 = $140

Therefore:

Savings = $60

Price paid = $140

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Comparing Sale Offers

Advertisements may present discounts in different ways.

For example:

Store A: 25% off

Store B: $20 off

Which gives the larger saving?

The answer depends on the original price.

Suppose the item costs:

$100

Store A:

25% of $100 = $25

Store B:

$20

So the savings are:

$25 versus $20

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What If the Original Price Is $60?

Store A:

25% of $60 = $15

Store B:

$20 off

Now the savings are:

$15 versus $20

The comparison has changed.

This demonstrates why a percentage discount cannot always be compared with a fixed discount without knowing the original price.

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Finding the Break-Even Price

Suppose one offer is:

20% off

and another is:

$15 off

At what original price are the savings equal?

Let the original price be:

P

Then:

20% of P = $15

0.20P = 15

Divide:

P = 15 ÷ 0.20

P = $75

At an original price of $75, both offers save:

$15

Below or above this price, the comparison changes.

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Comparing Discounts on Different Original Prices

Suppose:

Store A:

$80 shirt with 25% off

Store B:

$70 shirt with 15% off

Store A:

Discount:

$80 × 0.25 = $20

Final price:

$60

Store B:

Discount:

$70 × 0.15 = $10.50

Final price:

$59.50

A larger percentage discount does not automatically produce a lower final price because the original prices may differ.

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Multiple Discounts

Stores sometimes advertise offers such as:

20% off, then an additional 10% off

It may be tempting to say:

20% + 10% = 30%

But successive discounts are normally calculated one after another.

Suppose the original price is:

$100

First discount:

20% of $100 = $20

New price:

$80

Second discount:

10% of $80 = $8

Final price:

$72

Total savings:

$100 − $72 = $28

Therefore, the total effective discount is:

28%

not 30%.

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Why Successive Discounts Do Not Simply Add

After the first discount, the second percentage is applied to a smaller price.

For:

20% off, then 10% off

the multipliers are:

0.80 × 0.90

= 0.72

The customer pays:

72%

of the original price.

Therefore, the effective discount is:

28%

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Another Multiple-Discount Example

Original price:

$200

Sale:

30% off, then another 20% off

First discount:

$200 × 0.70 = $140

Second discount:

$140 × 0.80 = $112

Final price:

$112

Savings:

$200 − $112 = $88

Effective discount:

$88 / $200 × 100% = 44%

So:

30% off followed by 20% off = 44% effective discount

not 50%.

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Comparing One Discount with Multiple Discounts

Offer A:

40% off

Offer B:

25% off, then another 20% off

Suppose the original price is $100.

Offer A:

$100 × 0.60 = $60

Offer B:

$100 × 0.75 × 0.80 = $60

In this case, both produce the same final price.

The effective discount for Offer B is therefore also:

40%

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Buy One, Get One Free

Another common sale is:

Buy One, Get One Free

Suppose each item normally costs:

$30

Two items normally cost:

$60

Under the offer:

$30

is paid for two items.

Average cost per item:

$30 ÷ 2 = $15

Compared with buying two identical items at the normal price, this is equivalent to a:

50% reduction in the total price of the two-item purchase

provided the customer actually needs two items and both qualify.

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Buy Two, Get One Free

Suppose each item costs:

$12

Normally, three items cost:

3 × $12 = $36

Under:

Buy 2, Get 1 Free

the customer pays:

2 × $12 = $24

Savings:

$36 − $24 = $12

Percentage saving across the three-item bundle:

12 / 36 × 100%

= 33.3%

approximately.

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Percentage Off vs Extra Product

Suppose two packages normally contain 500 g.

Offer A:

20% off the price

Offer B:

20% extra product for the same price

These offers are not mathematically identical.

If the normal price is $10:

Offer A:

500 g for:

$8

Cost per 100 g:

$8 ÷ 5 = $1.60

Offer B:

20% extra of 500 g:

0.20 × 500 = 100 g

New quantity:

600 g

Cost per 100 g:

$10 ÷ 6 ≈ $1.67

Unit rates allow the offers to be compared consistently.

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Unit Price and Sales

A sale price does not automatically mean good value compared with other products.

Suppose:

Brand A:

750 g for $6 after discount.

Brand B:

1 kg for $7.50 at regular price.

Brand A:

$6 ÷ 0.75 = $8/kg

Brand B:

$7.50 ÷ 1 = $7.50/kg

The word sale alone does not tell us which option has the lower unit price.

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What Does "Good Value" Mean?

Evaluating value requires more than identifying the largest discount percentage.

Mathematical information that may matter includes:

• original price
• final price
• discount amount
• quantity received
• unit price
• number of items required
• additional fees
• total cost over time

Other practical considerations can include:

• quality
• durability
• usefulness
• whether the item is actually needed
• whether some of the quantity will be wasted

Mathematics helps provide the evidence needed to evaluate the options.

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Sale Price vs Value

Suppose:

Option A:

10 pens for $8

Option B:

20 pens for $14

Unit prices:

Option A:

$8 ÷ 10 = $0.80/pen

Option B:

$14 ÷ 20 = $0.70/pen

Option B has the lower unit price.

However, someone who needs only 5 pens would still have to spend:

$14

to receive that unit price.

"Better unit value" and "lower total spending" are not always the same thing.

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Discounts and Budgets

Suppose your budget is:

$100

An item originally costs:

$125

It is:

25% off

Discount:

$125 × 0.25 = $31.25

Final price:

$125 − $31.25 = $93.75

Money remaining:

$100 − $93.75 = $6.25

The sale brings the item within the $100 budget.

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What Percentage Was Saved?

Sometimes the original and sale prices are given instead of the discount percentage.

Suppose:

Original price:

$80

Sale price:

$60

Savings:

$80 − $60 = $20

Discount percentage:

$20 / $80 × 100%

= 25%

Therefore:

the discount was 25%

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Another Reverse Discount Problem

Original price:

$150

Sale price:

$105

Savings:

$150 − $105 = $45

Percentage discount:

45 / 150 × 100%

= 30%

Therefore:

30% off

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Finding the Original Price

Sometimes we know the final price and discount.

Suppose a jacket costs:

$72 after a 20% discount

A 20% discount means the customer pays:

80%

of the original price.

Let the original price be:

P

Then:

0.80P = 72

Therefore:

P = 72 ÷ 0.80

P = $90

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Another Original-Price Example

A device costs:

$195 after a 35% discount

Percentage paid:

65%

Therefore:

0.65P = 195

Calculate:

P = 195 ÷ 0.65

P = $300

Original price:

$300

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Discounts Followed by Additional Charges

A discount may not be the final calculation.

Suppose:

Original price:

$200

Discount:

25%

Discounted price:

$200 × 0.75 = $150

Suppose an illustrative 8% tax is then applied to the discounted price.

Tax:

$150 × 0.08 = $12

Final amount:

$150 + $12 = $162

The order of calculations matters.

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Coupons and Percentage Discounts

Suppose a $100 item has:

20% off

and then a:

$10 coupon

After the percentage discount:

$100 × 0.80 = $80

After the coupon:

$80 − $10 = $70

Final price:

$70

However, real stores may have rules about whether coupons can be combined or the order in which discounts apply.

Always use the conditions stated in the problem or offer.

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Comparing Three Sale Offers

Suppose an item normally costs:

$120

Offer A:

25% off

Final price:

$120 × 0.75 = $90

Offer B:

$35 off

Final price:

$120 − $35 = $85

Offer C:

10% off, then 15% off

First:

$120 × 0.90 = $108

Then:

$108 × 0.85 = $91.80

The final prices are:

A comparison table makes the differences much easier to see.

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Reading Sale Information from Tables

Suppose a store provides:

To interpret the table, we might calculate both the savings and final prices.

Shoes:

Savings = $20

Final = $60

Jacket:

Savings = $36

Final = $84

Backpack:

Savings = $9

Final = $51

Headphones:

Savings = $60

Final = $90

Tables allow several financial options to be compared systematically.

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Interpreting Sale Graphs

A graph might show:

• original prices
• sale prices
• discount percentages
• savings
• prices at different stores

When reading a financial graph:

1. Read the title.

2. Check the axes.

3. Check the units.

4. Identify whether values show prices, percentages, or savings.

5. Compare actual numerical values rather than relying only on visual appearance.

6. Check whether the graph scale could exaggerate differences.

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Sale Advertisements

Advertisements commonly use phrases such as:

UP TO 70% OFF

SAVE $50

FROM $19.99

BUY ONE, GET ONE FREE

20% EXTRA

These statements do not all mean the same thing.

For example:

"Up to 70% off"

means some qualifying items may have a 70% discount. It does not mean every item is discounted by 70%.

Careful interpretation is part of mathematical financial literacy.

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Percentage Discount vs Dollar Savings

A large percentage discount does not necessarily mean a large amount of money saved.

Item A:

$20 with 50% off

Savings:

$10

Item B:

$500 with 10% off

Savings:

$50

Item A has the larger percentage discount.

Item B produces the larger dollar saving.

Which measure matters depends on the question being asked.

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Original Price Matters

Compare:

50% off $20

and:

20% off $100

First:

0.50 × $20 = $10 saved

Second:

0.20 × $100 = $20 saved

The smaller percentage produces the larger dollar saving because it applies to a larger original price.

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Estimating Sale Prices

Estimation can help check calculations.

Suppose:

32% off $198

Estimate:

About 30% of $200:

≈ $60

So the sale price should be roughly:

$140

Exact discount:

$198 × 0.32 = $63.36

Exact sale price:

$198 − $63.36 = $134.64

The exact answer is reasonably close to the estimate.

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Checking Whether an Answer Is Reasonable

If an item is:

20% off

the final price should be:

• lower than the original price
• greater than 50% of the original price
• exactly 80% of the original price

For a $100 item:

A final price of:

$80

makes sense.

A final price of:

$20

would mean the discount amount has been confused with the sale price.

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Worked Example 1: Basic Discount

Original price:

$70

Discount:

20%

Savings:

$70 × 0.20 = $14

Final price:

$70 − $14 = $56

Answer:

Savings = $14

Final price = $56

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Worked Example 2: 35% Off

Original price:

$160

Discount:

35%

Savings:

$160 × 0.35 = $56

Final price:

$160 − $56 = $104

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Worked Example 3: Using a Multiplier

Original price:

$280

Discount:

15%

Percentage paid:

85%

Calculate:

$280 × 0.85 = $238

Final price:

$238

Savings:

$42

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Worked Example 4: Comparing Offers

A $200 item has two possible offers.

Offer A:

30% off

Savings:

$60

Final:

$140

Offer B:

$50 off

Savings:

$50

Final:

$150

The offers produce final prices of:

$140 and $150

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Worked Example 5: Successive Discounts

Original price:

$250

Discounts:

20% off, then 15% off

First:

$250 × 0.80 = $200

Second:

$200 × 0.85 = $170

Final price:

$170

Savings:

$250 − $170 = $80

Effective discount:

80 / 250 × 100% = 32%

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Worked Example 6: Find the Discount Percentage

Original price:

$240

Sale price:

$180

Savings:

$240 − $180 = $60

Percentage discount:

60 / 240 × 100%

= 25%

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Worked Example 7: Find the Original Price

Sale price:

$126

Discount:

30%

Percentage paid:

70%

Therefore:

0.70P = 126

P = 126 ÷ 0.70

P = $180

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Worked Example 8: Buy Two, Get One Free

Each shirt costs:

$24

Normally three shirts cost:

3 × $24 = $72

Under the offer:

2 × $24 = $48

Savings:

$24

Percentage saving:

24 / 72 × 100%

≈ 33.3%

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Worked Example 9: Unit-Price Comparison

Offer A:

600 g for $4.80

Offer B:

900 g for $6.75

Offer A:

$4.80 ÷ 6 = $0.80 per 100 g

Offer B:

$6.75 ÷ 9 = $0.75 per 100 g

The unit prices provide a common basis for comparing the offers.

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Worked Example 10: Budget and Sale

Budget:

$150

Original price:

$180

Discount:

25%

Final price:

$180 × 0.75 = $135

Money remaining:

$150 − $135 = $15

The discounted item fits within the stated budget.

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Evaluating a Sale

When deciding whether a sale represents useful value, ask:

What is the original price?

What percentage is actually discounted?

How many dollars will I save?

What is the final price?

How does the unit price compare with alternatives?

Are there additional charges?

Do I need to buy more items to receive the discount?

Will I actually use the extra quantity?

Is another store's regular price lower?

Does the purchase fit the available budget?

A large discount percentage by itself does not answer all of these questions.

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A Reliable Discount Strategy

Step 1: Identify the original price.

Step 2: Identify the discount percentage or offer.

Step 3: Convert the percentage to a decimal or fraction.

Step 4: Calculate the discount amount.

Step 5: Subtract the discount from the original price.

Step 6: Include any additional calculations stated in the problem.

Step 7: If comparing offers, calculate each final price on the same basis.

Step 8: Compare savings and unit prices when relevant.

Step 9: Estimate to check your calculation.

Step 10: Interpret the numbers in the context of the purchase.

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Common Mistakes

Mistake 1: Treating the discount as the final price

20% off $100 means:

$20 saved

not:

$20 final price

The final price is:

$80

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Mistake 2: Forgetting to convert the percentage

Incorrect:

$80 × 25

Correct:

$80 × 0.25

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Mistake 3: Subtracting the percentage number directly

Incorrect:

$100 − 20 = $80

This happens to produce the correct number only because the original price is $100.

For $200:

20% off is:

$40

not $20.

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Mistake 4: Adding successive discounts

20% off followed by 10% off is:

28% effective discount

not 30%.

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Mistake 5: Assuming the largest percentage means the lowest price

Original prices may be different.

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Mistake 6: Assuming a sale automatically means good value

Another product may have a lower regular price or lower unit price.

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Mistake 7: Ignoring conditions

Some discounts require:

• buying multiple items
• spending a minimum amount
• membership
• using a coupon
• purchasing during a particular period

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Mistake 8: Ignoring how much is actually needed

A lower unit price may require purchasing much more than necessary.

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Did You Know?

Discount mathematics combines several mathematical ideas.

When comparing sales, we may use:

• percentages
• decimals
• fractions
• ratios
• unit rates
• proportions
• estimation
• tables
• graphs
• algebra

This is why shopping provides a useful real-world application of mathematics.

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Key Terms

• Original price: Price before a discount.
• Discount: Reduction in price.
• Discount rate: Percentage by which the original price is reduced.
• Discount amount: Amount of money removed from the original price.
• Sale price: Price after the discount.
• Final price: Amount paid after the calculations specified in the problem.
• Savings: Difference between the original price and the discounted price.
• Multiplier: Decimal used to calculate a percentage of an amount directly.
• Successive discount: One discount applied after another.
• Effective discount: Overall percentage reduction produced by one or more discounts.
• Unit price: Cost per one item or standard unit of quantity.
• Coupon: Offer that reduces a price when its conditions are satisfied.
• Break-even point: Value at which two financial options produce the same result.
• Value: Relationship between what is received and what it costs.

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Key Equations

Discount amount:

Discount = Original Price × Discount Rate

Sale price:

Sale Price = Original Price − Discount

Percentage paid:

Percentage Paid = 100% − Discount Percentage

Multiplier method:

Sale Price = Original Price × Percentage Paid as a Decimal

Savings:

Savings = Original Price − Sale Price

Discount percentage:

Discount % = Savings / Original Price × 100%

Original price:

Original Price = Sale Price / Percentage Paid as a Decimal

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Key Takeaways

• A discount reduces the original price of a product or service.
• Percentage discounts are calculated from the original price unless the offer specifies otherwise.
• The discount amount is the amount saved, not usually the final price.
• The sale price equals the original price minus the discount.
• A multiplier can calculate a sale price directly.
• A 20% discount means paying 80% of the original price.
• Savings can be calculated in dollars or as a percentage.
• Fixed-dollar discounts and percentage discounts should be compared using actual savings or final prices.
• Different original prices can make percentage comparisons misleading.
• Successive discounts are applied one after another and normally should not simply be added.
• Unit prices are useful when sale packages contain different quantities.
• "Buy one, get one free" and similar offers should be analyzed using the total cost and quantity received.
• The largest advertised discount does not automatically produce the lowest final price.
• A lower unit price does not automatically mean lower total spending.
• Additional fees or charges may affect the final amount.
• Estimation is useful for checking discount calculations.
• Sale advertisements should be interpreted carefully, especially phrases such as "up to", "from", and "extra."
• Evaluating value requires comparing the numerical evidence with the quantity, cost, conditions, and purpose of the purchase.
• The most useful question is not simply "How big is the discount?" but:

"What will I actually pay, what will I receive, and how does that compare with the alternatives?"

3. Interest and Growth

4. Probability as Fractions and Percentages

What Is Probability?

Probability is a measure of how likely an event is to happen.

We use probability whenever there is uncertainty about an outcome.

Examples include:

• whether a coin lands heads or tails
• which number appears when a die is rolled
• which color is selected from a bag
• whether it will rain
• whether a machine produces a defective item
• how often a particular result may occur in repeated trials

Probability does not usually tell us exactly what will happen next. Instead, it describes how likely different outcomes are.

---

The Probability Scale

Probability ranges from:

0 to 1

or:

0% to 100%

Important points include:

0 = 0% = impossible

1/4 = 0.25 = 25% = unlikely

1/2 = 0.5 = 50% = even chance

3/4 = 0.75 = 75% = likely

1 = 100% = certain

A probability can never be less than 0 or greater than 1.

---

Probability as a Fraction

When outcomes are equally likely:

Probability = Number of favorable outcomes / Total number of possible outcomes

We often write this as:

P(event) = favorable outcomes / total outcomes

For example, a fair six-sided die has six possible outcomes:

1, 2, 3, 4, 5, 6

The probability of rolling a 4 is:

P(4) = 1/6

There is one favorable outcome out of six possible outcomes.

---

Probability as a Decimal

A fraction can be converted to a decimal by dividing:

numerator ÷ denominator

For example:

1/4 = 1 ÷ 4 = 0.25

Therefore:

P = 0.25

---

Probability as a Percentage

To convert a decimal probability into a percentage:

Decimal × 100%

For example:

0.25 × 100% = 25%

Therefore:

1/4 = 0.25 = 25%

All three values describe exactly the same probability.

---

Common Probability Equivalents

Recognizing these common equivalents makes probability comparisons much easier.

---

Simple Probability

Suppose a bag contains:

3 red counters

2 blue counters

5 counters altogether

What is the probability of selecting red?

Favorable outcomes:

3

Total outcomes:

5

Therefore:

P(red) = 3/5

Convert to decimal:

3 ÷ 5 = 0.6

Convert to percentage:

0.6 × 100% = 60%

Therefore:

P(red) = 3/5 = 0.6 = 60%

---

Probability with a Fair Coin

A fair coin has two possible outcomes:

Heads

Tails

Therefore:

P(heads) = 1/2

and:

P(tails) = 1/2

Convert:

1/2 = 0.5 = 50%

So each outcome has a 50% probability on a single fair toss.

---

Probability with a Fair Die

A standard fair die has six possible outcomes:

1, 2, 3, 4, 5, 6

Each individual number has probability:

1/6

What is the probability of rolling an even number?

Even numbers:

2, 4, 6

There are:

3 favorable outcomes

Therefore:

P(even) = 3/6

Simplify:

P(even) = 1/2

So:

P(even) = 0.5 = 50%

---

Another Die Example

What is the probability of rolling a number greater than 4?

Possible favorable outcomes:

5, 6

Therefore:

P(number > 4) = 2/6

Simplify:

1/3

Decimal:

1 ÷ 3 ≈ 0.333

Percentage:

≈ 33.3%

---

Probability with a Spinner

Suppose a spinner has:

8 equal sections

with:

3 red

2 blue

2 green

1 yellow

Probability of red:

3/8

Decimal:

0.375

Percentage:

37.5%

Probability of yellow:

1/8

Decimal:

0.125

Percentage:

12.5%

Because red occupies more equal sections than yellow, red is more likely to occur.

---

Equally Likely Outcomes

The simple formula:

P(event) = favorable outcomes / total outcomes

works directly when the individual outcomes are equally likely.

For example, with a fair die, each face has the same probability.

But suppose a spinner contains sections of different sizes.

A large section may be more likely than a small section.

In that situation, simply counting the number of sections may not correctly determine probability.

The size or probability of each outcome must also be considered.

---

Favorable Outcomes

A favorable outcome is an outcome that satisfies the event we are interested in.

Suppose a die is rolled.

Event:

Roll a number less than 5

Favorable outcomes:

1, 2, 3, 4

Therefore:

P(number < 5) = 4/6

Simplify:

2/3

This does not mean that 1, 2, 3, and 4 are "good." In probability, favorable simply means that the outcome matches the event being investigated.

---

Impossible Events

An impossible event has probability:

0

For example, rolling a 9 on a standard six-sided die is impossible.

Therefore:

P(9) = 0

or:

0%

---

Certain Events

A certain event has probability:

1

For example, rolling a number less than 7 on a standard six-sided die is certain.

Possible outcomes:

1, 2, 3, 4, 5, 6

All six satisfy the condition.

Therefore:

P(number < 7) = 6/6 = 1 = 100%

---

Likelihood Language

Probability can also be described using words.

Impossible: probability = 0

Unlikely: probability closer to 0 than 0.5

Even chance: probability = 0.5

Likely: probability closer to 1 than 0.5

Certain: probability = 1

These words describe general levels of likelihood rather than always specifying an exact probability.

---

Comparing Probabilities

Suppose:

Event A:

P(A) = 1/4

Event B:

P(B) = 0.40

Event C:

P(C) = 35%

Convert them to the same form.

Event A:

1/4 = 25%

Event B:

0.40 = 40%

Event C:

35%

Now the probabilities are:

A = 25%

B = 40%

C = 35%

Putting probabilities in the same form makes them much easier to compare.

---

Comparing Fractions

Suppose:

P(A) = 2/5

and:

P(B) = 3/8

Convert to decimals:

2/5 = 0.40

3/8 = 0.375

Therefore:

0.40 > 0.375

So event A has the greater probability.

---

Complementary Events

The complement of an event is the event not happening.

For any event A:

P(A) + P(not A) = 1

or:

100%

Suppose:

P(rain) = 30%

Then:

P(no rain) = 100% − 30%

= 70%

---

Complement Example with a Die

What is the probability of not rolling a 6?

Probability of rolling 6:

1/6

Therefore:

P(not 6) = 1 − 1/6

= 5/6

or approximately:

83.3%

---

Theoretical Probability

Theoretical probability is based on a mathematical model of possible outcomes.

For a fair coin:

P(heads) = 1/2

For a fair six-sided die:

P(6) = 1/6

These probabilities can be calculated before conducting an experiment.

---

Experimental Probability

Experimental probability is based on actual observations or trials.

The equation is:

Experimental Probability = Number of Times Event Occurs / Total Number of Trials

Suppose a coin is tossed 50 times.

Heads occurs:

27 times

Experimental probability:

27/50

= 0.54

= 54%

The theoretical probability is 50%, but an experiment does not have to produce exactly 50%.

---

Why Experimental Results Vary

Random processes naturally produce variation.

A fair coin tossed 10 times might produce:

7 heads and 3 tails

That does not necessarily mean:

P(heads) = 70%

The theoretical probability remains:

50%

for a fair coin.

With a larger number of trials, experimental proportions often become closer to the theoretical probability.

This idea is connected to the law of large numbers.

---

Relative Frequency

Experimental probability is also called relative frequency.

Suppose a basketball player takes:

80 free throws

and makes:

60

Relative frequency of success:

60/80

= 3/4

= 0.75

= 75%

This information can be used to describe past performance and may help estimate future outcomes, while recognizing that future results are uncertain.

---

Using Probability to Make Predictions

Probability can help estimate how many times an event may occur over many trials.

Suppose:

P(red) = 0.30

and a random selection experiment is performed:

200 times

Expected number of red outcomes:

0.30 × 200 = 60

Therefore, we might predict:

about 60 red outcomes

This is a prediction, not a guarantee.

---

Expected Frequency

The expected number of times an event occurs can be calculated using:

Expected Frequency = Probability × Number of Trials

For example:

Probability of success:

0.4

Number of trials:

150

Expected frequency:

0.4 × 150 = 60

We would expect approximately:

60 successes

over many similar trials.

---

Prediction with Fractions

Suppose:

P(blue) = 3/5

and there will be:

100 selections

Expected blue outcomes:

3/5 × 100

= 60

Prediction:

about 60 blue outcomes

---

Prediction with Percentages

Suppose a machine historically produces:

2% defective items

If:

5,000 items

are produced under similar conditions, a simple prediction based on that rate is:

2% of 5,000

0.02 × 5000 = 100

So the expected number is:

about 100 defective items

Actual results could be higher or lower.

---

Probability in Weather Information

Weather forecasts often use probabilities.

Suppose a forecast reports:

70% chance of rain

This describes a probability associated with the forecast event and its specified place and time period.

It does not mean that it will rain for exactly 70% of the day.

It also does not guarantee that rain will occur.

A probability describes uncertainty, not certainty.

---

Probability in Sports

Sports statistics often use past frequencies.

Suppose a player successfully completes:

42 of 60 attempts

Success rate:

42/60

Simplify:

7/10

Decimal:

0.70

Percentage:

70%

If the player makes another 100 attempts under broadly similar conditions, 70 successes might be used as a simple prediction based on the historical rate.

But actual performance can change.

---

Probability in Games

Games frequently involve probability.

Examples include:

• dice
• cards
• spinners
• coins
• random number generators

Suppose a game awards a prize if a player rolls a 6.

Probability of winning:

1/6

Probability of not winning:

5/6

Knowing these probabilities can help us understand how the game is structured.

---

Probability with Cards

A standard deck contains:

52 cards

There are:

4 suits

with:

13 cards in each suit

Probability of drawing a heart:

13/52

Simplify:

1/4

Therefore:

P = 0.25 = 25%

---

Another Card Example

There are four aces in a standard 52-card deck.

Therefore:

P(ace) = 4/52

Simplify:

1/13

Decimal:

≈ 0.077

Percentage:

≈ 7.7%

---

Probability from Tables

Suppose a survey records preferred transportation:

Total:

24 + 36 + 20 + 20 = 100

If one student is randomly selected:

P(bus) = 36/100 = 0.36 = 36%

P(bicycle) = 20/100 = 0.20 = 20%

Tables can therefore be used to calculate probabilities from data.

---

Probability from Graphs

Graphs can also provide information used to estimate probabilities.

Suppose a graph shows:

Red = 40 outcomes

Blue = 30 outcomes

Green = 20 outcomes

Yellow = 10 outcomes

Total:

100 outcomes

Therefore:

P(red) = 40/100 = 40%

P(blue) = 30/100 = 30%

P(green) = 20/100 = 20%

P(yellow) = 10/100 = 10%

---

Interpreting Probability Data Carefully

Suppose:

20 people are surveyed.

16 choose option A.

Experimental proportion:

16/20 = 80%

It would be correct to say:

80% of the surveyed group chose A.

It would require additional assumptions to claim that exactly 80% of a much larger population would choose A.

The size and selection of the sample matter when probability and statistics are used to make predictions.

---

Probability Does Not Guarantee Individual Outcomes

Suppose:

P(win) = 90%

This means winning is very likely under the stated model.

It does not mean winning is certain.

A 10% probability of not winning still exists.

Similarly:

P(event) = 1%

does not mean the event is impossible.

Low probability and impossibility are different ideas.

---

Random Does Not Mean "Equal"

A process can be random without every outcome having the same probability.

For example, imagine a spinner where:

70% of the area is blue

and:

30% is red.

The result may still be random, but:

P(blue) = 0.70

and:

P(red) = 0.30

---

Independent Repeated Events

Suppose a fair coin has landed heads five times in a row.

If each toss is independent, the probability of heads on the next toss is still:

1/2

The coin does not "owe" us a tail.

Previous independent results do not change the probability of the next toss.

This common misunderstanding is sometimes called the gambler's fallacy.

---

Probability and Risk

Probability is often used to describe risk.

For example, suppose two hypothetical events have probabilities:

Event A:

2%

Event B:

20%

Event B is ten times as likely under the stated model:

20% ÷ 2% = 10

However, probability is only one part of evaluating risk. The consequences of an event may also matter.

---

Worked Example 1: Colored Counters

A bag contains:

4 red

5 blue

1 yellow

Total:

10

Probability of blue:

5/10 = 1/2

Decimal:

0.5

Percentage:

50%

---

Worked Example 2: Die

What is the probability of rolling a number greater than 2?

Favorable outcomes:

3, 4, 5, 6

Therefore:

P(>2) = 4/6

Simplify:

2/3

Decimal:

≈ 0.667

Percentage:

≈ 66.7%

---

Worked Example 3: Spinner

A spinner has 20 equal sections.

Eight are green.

Probability:

8/20

Simplify:

2/5

Decimal:

0.4

Percentage:

40%

---

Worked Example 4: Compare Probabilities

Event A:

3/4

Event B:

0.70

Event C:

72%

Convert:

A = 75%

B = 70%

C = 72%

So their probabilities are:

75%, 70%, and 72%

Converting to a common representation makes the comparison clear.

---

Worked Example 5: Complement

Probability of success:

0.65

Probability of failure:

1 − 0.65 = 0.35

Therefore:

P(failure) = 35%

---

Worked Example 6: Experimental Probability

A spinner is used:

200 times

Blue occurs:

46 times

Experimental probability:

46/200

= 0.23

= 23%

---

Worked Example 7: Prediction

Suppose:

P(success) = 35%

There will be:

400 trials

Expected successes:

0.35 × 400 = 140

Prediction:

about 140 successes

---

Worked Example 8: Using Historical Data

A sports player succeeds:

72 times in 90 attempts

Experimental probability:

72/90

= 0.8

= 80%

If the player makes another 50 attempts under similar conditions:

Expected successes:

0.80 × 50 = 40

A simple prediction is:

about 40 successes

This remains an estimate rather than a guarantee.

---

Worked Example 9: Probability from a Table

A survey gives:

Total:

100

If one response is randomly selected:

P(A) = 45%

P(B) = 30%

P(C) = 15%

P(D) = 10%

---

Worked Example 10: Multi-Step Probability

A bag contains:

6 red

3 blue

1 green

Total:

10

Probability of selecting something that is not red:

Blue or green outcomes:

3 + 1 = 4

Therefore:

P(not red) = 4/10

= 2/5

= 0.4

= 40%

We could also use the complement:

P(red) = 6/10 = 60%

Therefore:

P(not red) = 100% − 60% = 40%

---

Making Predictions from Probability

Probability-based predictions are most useful over many trials.

Suppose:

P(A) = 0.25

For 20 trials:

Expected frequency:

0.25 × 20 = 5

For 100 trials:

0.25 × 100 = 25

For 1,000 trials:

0.25 × 1000 = 250

The actual results may differ from these numbers, especially for smaller numbers of trials.

---

A Reliable Probability Strategy

Step 1: Identify the event.

What exactly are you trying to find?

Step 2: Identify all possible outcomes.

Step 3: Determine whether the outcomes are equally likely.

Step 4: Count the favorable outcomes.

Step 5: Calculate:

P(event) = favorable outcomes / total outcomes

Step 6: Simplify the fraction if possible.

Step 7: Convert to a decimal or percentage if required.

Step 8: Check that the probability is between 0 and 1.

Step 9: Interpret what the probability means in context.

Step 10: If making a prediction, remember that expected results are not guaranteed results.

---

Common Mistakes

Mistake 1: Writing a probability greater than 1

For example:

P(A) = 1.4

cannot be a valid probability.

---

Mistake 2: Forgetting the total number of outcomes

If a bag contains 3 red and 7 blue counters:

P(red) = 3/10

not:

3/7

---

Mistake 3: Confusing percentages and decimals

25% = 0.25

not:

25

---

Mistake 4: Assuming all outcomes are equally likely

This is only valid when the model supports that assumption.

---

Mistake 5: Treating a prediction as a guarantee

A predicted 60 successes does not mean exactly 60 must occur.

---

Mistake 6: Treating unlikely as impossible

A probability of 1% is small, but it is not zero.

---

Mistake 7: Treating likely as certain

A probability of 90% is high, but it is not 100%.

---

Mistake 8: Assuming short experiments must match theoretical probability

Ten fair coin tosses do not have to produce exactly five heads.

---

Mistake 9: Assuming previous independent results control the next result

Five heads in a row do not make tails "due" on the next independent toss.

---

Did You Know?

Probability connects many areas of mathematics and real-world decision-making.

Probability is used in:

• weather forecasting
• science
• genetics
• medicine and public health research
• engineering
• manufacturing
• sports statistics
• insurance
• economics
• quality control
• computer science
• games

In each case, probability helps describe uncertainty.

---

Key Terms

• Probability: Measure of how likely an event is.
• Outcome: One possible result.
• Event: One outcome or a collection of outcomes being investigated.
• Favorable outcome: Outcome that satisfies the event being studied.
• Equally likely: Outcomes having the same probability.
• Impossible event: Event with probability 0.
• Certain event: Event with probability 1.
• Likelihood: How probable an event is.
• Complement: Event consisting of an event not occurring.
• Theoretical probability: Probability calculated from a mathematical model.
• Experimental probability: Probability estimated from observed results.
• Relative frequency: Proportion of trials in which an event occurs.
• Trial: One performance of a probability experiment.
• Expected frequency: Predicted number of occurrences based on probability.
• Random: Involving uncertainty in individual outcomes.
• Independent events: Events where the outcome of one does not affect the probability of another.
• Prediction: Estimate of future results based on available information.

---

Key Equations and Rules

Simple probability for equally likely outcomes:

P(event) = Favorable Outcomes / Total Possible Outcomes

Experimental probability:

P(event) = Number of Times Event Occurs / Total Number of Trials

Complement rule:

P(A) + P(not A) = 1

Therefore:

P(not A) = 1 − P(A)

Expected frequency:

Expected Frequency = Probability × Number of Trials

Fraction to decimal:

Numerator ÷ Denominator

Decimal to percentage:

Decimal × 100%

Percentage to decimal:

Percentage ÷ 100

---

Key Takeaways

• Probability measures how likely an event is to occur.
• Probabilities range from 0 to 1, or from 0% to 100%.
• A probability of 0 represents an impossible event.
• A probability of 1 represents a certain event.
• An even chance has probability 1/2 = 0.5 = 50%.
• Probabilities can be written as fractions, decimals, or percentages.
• Different forms can represent exactly the same probability.
• For equally likely outcomes, probability can be calculated using favorable outcomes divided by total possible outcomes.
• Probabilities are easier to compare when converted to the same form.
• Complementary probabilities add to 1, or 100%.
• Theoretical probability comes from a mathematical model.
• Experimental probability comes from observed data.
• Experimental results may differ from theoretical probabilities.
• Larger numbers of trials often produce experimental proportions closer to theoretical probabilities.
• Expected frequency can be calculated by multiplying probability by the number of trials.
• Probability can be used to make predictions, but predictions are not guarantees.
• Random does not necessarily mean that all outcomes are equally likely.
• Past independent outcomes do not make a particular future outcome "due."
• Real-world probability information should be interpreted in context, especially when it comes from samples or historical data.
• Probability is useful because it gives us a mathematical way to describe and reason about uncertainty.

5. Review and Real-World Projects