What Is Probability?

Every day, we make predictions about what might happen:

• Will it rain tomorrow?
• Will a coin land on heads?
• Will a basketball player make the next shot?
• Will a bus arrive on time?
• Will a six-sided die roll a 6?

Some events are very predictable. Others are uncertain.

Probability is the branch of mathematics that helps us describe and measure this uncertainty.

Probability tells us how likely an event is to happen.

For example:

• The probability that the Sun will rise tomorrow is extremely close to certain.
• The probability of rolling a 7 on a normal six-sided die is impossible.
• The probability of getting heads when flipping a fair coin is 1/2.
• The probability of rain tomorrow may be estimated from weather data and forecasting models.

Probability allows us to move beyond words such as "maybe" or "probably" and describe likelihood more precisely.

What Is Probability?

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

Probability can be expressed as:

• a fraction
• a decimal
• a percentage

For example:

1/2 = 0.5 = 50%

These all represent the same probability.

A probability must lie between:

0 and 1

or equivalently:

0% and 100%

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The Probability Scale

A probability scale runs from:

0 to 1

The important points are:

0 → impossible

0.5 → equally likely to happen or not happen

1 → certain

Values between these points describe different levels of likelihood.

For example:

0.1 → very unlikely

0.3 → unlikely

0.5 → even chance

0.7 → likely

0.9 → very likely

The closer a probability is to 1, the more likely the event is.

The closer it is to 0, the less likely the event is.

Impossible Events

An impossible event cannot happen under the stated conditions.

Its probability is:

P(event) = 0

Examples include:

• rolling a 9 on a standard six-sided die
• choosing February 30 from a normal calendar
• flipping a coin and having it land as a standard die showing 4
• selecting a blue marble from a bag containing only red marbles

An impossible event has a probability of:

0 = 0%

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Certain Events

A certain event must happen under the stated conditions.

Its probability is:

P(event) = 1

For example, suppose a bag contains:

5 red marbles and no other colours

If you randomly select one marble:

P(red) = 1

Selecting a red marble is certain.

Another example is rolling a standard six-sided die and getting a number less than 7.

Possible results are:

1, 2, 3, 4, 5, 6

Every possible result is less than 7.

Therefore:

P(number less than 7) = 1

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Likely Events

A likely event has a relatively high probability of occurring, but it is not certain.

For example, suppose a bag contains:

8 blue marbles

and

2 yellow marbles

There are 10 marbles altogether.

Selecting blue is likely because most of the marbles are blue.

P(blue) = 8/10

P(blue) = 0.8

P(blue) = 80%

An 80% probability means the event is likely—but it can still fail to happen.

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Unlikely Events

An unlikely event has a relatively low probability of occurring, but it is still possible.

Using the same bag:

8 blue marbles
2 yellow marbles

P(yellow) = 2/10

= 0.2

= 20%

Selecting yellow is unlikely.

But it is not impossible.

There are yellow marbles in the bag, so the event can occur.

---

Even Chance

An event has an even chance when it is equally likely to happen or not happen.

Its probability is:

1/2 = 0.5 = 50%

A common example is flipping a fair coin.

Possible outcomes:

Heads

Tails

If the coin is fair:

P(heads) = 1/2

P(tails) = 1/2

Both outcomes have an equal probability.

Probability as a Fraction

For simple situations where all outcomes are equally likely:

Probability = number of favourable outcomes / total number of possible outcomes

For example, consider a standard six-sided die.

Possible outcomes:

1, 2, 3, 4, 5, 6

Suppose we want:

P(rolling a 4)

There is one favourable outcome:

4

There are six possible outcomes.

Therefore:

P(4) = 1/6

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Worked Example: Rolling an Even Number

A standard die has:

1, 2, 3, 4, 5, 6

The even numbers are:

2, 4, 6

There are:

3 favourable outcomes

out of:

6 possible outcomes

Therefore:

P(even) = 3/6

= 1/2

= 0.5

= 50%

Rolling an even number has an even chance.

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Worked Example: Rolling More Than 4

Possible outcomes:

1, 2, 3, 4, 5, 6

Numbers greater than 4:

5, 6

Therefore:

P(number greater than 4) = 2/6

= 1/3

≈ 0.333

≈ 33.3%

This event is possible but less likely than not.

Probability as a Decimal

Probabilities can also be written as decimals.

For example:

1/4 = 0.25

1/2 = 0.5

3/4 = 0.75

1 = 1.0

The decimal still lies between:

0 and 1

For example:

P(rain) = 0.75

means there is a 75% probability of rain.

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Probability as a Percentage

Probabilities are often communicated as percentages.

For example:

0.2 = 20%

0.45 = 45%

0.5 = 50%

0.8 = 80%

1 = 100%

To convert a decimal probability to a percentage:

Multiply by 100

For example:

0.68 × 100 = 68%

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Fractions, Decimals, and Percentages

The same probability can be written in several forms.

It is useful to move easily between these forms because probability information may be presented differently in different situations.

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Comparing Probabilities

Probability allows us to compare events mathematically.

Suppose:

P(A) = 0.25

P(B) = 0.70

Since:

0.70 > 0.25

Event B is more likely than Event A.

We do not need to know what the events actually are to make this comparison.

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Worked Example: Comparing Events

Suppose:

P(rain) = 30%

P(sunshine) = 65%

P(snow) = 5%

We can compare:

65% > 30% > 5%

Therefore:

• sunshine is the most likely of these events
• rain is less likely than sunshine
• snow is the least likely

This does not mean sunshine is guaranteed.

A 65% probability still leaves a 35% chance that the event will not occur.

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Probability Does Not Predict Exactly What Will Happen

This is one of the most important ideas in probability.

Suppose:

P(heads) = 1/2

That does not mean every two coin flips must produce:

one head and one tail

You could flip:

heads, heads

or:

tails, tails

Probability describes likelihood, not a guaranteed short-term pattern.

Over many repeated trials, results often become more consistent with the expected probabilities.

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A 90% Probability Is Not a Guarantee

Suppose the probability of an event is:

90%

This means the event is very likely.

But there is still:

10%

probability that it will not occur.

Similarly:

1% probability

does not mean impossible.

An unlikely event can still happen.

The distinction between:

unlikely

and

impossible

is extremely important.

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Probability and Weather Forecasts

Weather forecasts frequently use probabilities.

Suppose a forecast reports:

70% chance of rain

This indicates that rain is considered likely under the forecasting method and conditions.

It does not mean:

• it will rain for 70% of the day
• 70% of the city must receive rain
• rain is guaranteed

Probability describes the estimated likelihood of the defined event.

Probability in Sports

Probability is widely used in sports.

It can help describe:

• probability of making a shot
• probability of scoring a goal
• probability of winning a point
• probability of a particular result based on historical data

Suppose a basketball player has made:

80 of 100 free throws

Based only on these previous results, we might estimate the player's probability of making a similar future free throw as approximately:

80/100 = 0.8

or:

80%

However, this is an estimate based on previous observations.

The next shot is not guaranteed to go in.

---

Probability in Games

Many games depend heavily on probability.

Examples include:

• dice
• cards
• spinners
• coins
• numbered balls

Suppose a spinner has four equal sections:

Red, Blue, Green, Yellow

If each section is the same size:

P(red) = 1/4

P(blue) = 1/4

P(green) = 1/4

P(yellow) = 1/4

Each outcome is equally likely.

Equally Likely Outcomes

Two or more outcomes are equally likely when they have the same probability.

For a fair coin:

P(heads) = P(tails)

For a fair six-sided die:

P(1) = P(2) = P(3) = P(4) = P(5) = P(6)

Each individual die result has probability:

1/6

But groups of outcomes may have different probabilities.

For example:

P(rolling 1) = 1/6

P(rolling an even number) = 3/6

Therefore, rolling an even number is more likely than rolling exactly 1.

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Sample Space

The sample space is the set of all possible outcomes of an experiment.

For a coin:

Sample space:

{Heads, Tails}

For a six-sided die:

Sample space:

{1, 2, 3, 4, 5, 6}

For a spinner labelled A, B, C and D:

Sample space:

{A, B, C, D}

Identifying the sample space helps us calculate probability correctly.

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Outcome and Event

An outcome is one possible result.

For example, rolling:

4

is an outcome.

An event can contain one or more outcomes.

For example:

rolling an even number

contains the outcomes:

2, 4, 6

Therefore:

P(even) = 3/6

= 1/2

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Complementary Events

If an event either happens or does not happen, the probabilities must total:

1

For example:

P(rain) = 0.7

Therefore:

P(no rain) = 1 − 0.7

= 0.3

Or:

70% + 30% = 100%

These are called complementary events.

Another example:

P(win) = 0.42

P(not win) = 1 − 0.42

= 0.58

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Probabilities Must Make Sense

Because probability must be between 0 and 1, the following are possible:

0

0.17

0.5

0.93

1

But these cannot be valid probabilities:

−0.4

1.2

145%

A probability cannot be:

• less than 0
• greater than 1
• less than 0%
• greater than 100%

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Theoretical Probability

Theoretical probability is calculated from the possible outcomes when those outcomes are known and equally likely.

For example, on a fair six-sided die:

P(6) = 1/6

This can be calculated without actually rolling the die.

Similarly:

P(even number) = 3/6

= 1/2

This is based on the mathematical structure of the situation.

---

Experimental Probability

Sometimes probability is estimated using observations.

Suppose a thumbtack is dropped 100 times.

It lands point-up 37 times.

Experimental probability:

P(point-up) = number of times event occurred / total number of trials

P(point-up) = 37/100

= 0.37

= 37%

This is called experimental probability or empirical probability.

Theoretical and Experimental Probability May Differ

Suppose you flip a fair coin 10 times.

The theoretical probability of heads is:

1/2

Therefore, you might expect about five heads.

But suppose your results are:

7 heads and 3 tails

Experimental probability of heads:

7/10 = 0.7

Theoretical probability:

1/2 = 0.5

These are different.

That does not necessarily mean the coin is unfair.

Random variation can produce different results, especially with a small number of trials.

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Probability Over Many Trials

Suppose we flip a fair coin repeatedly.

After 10 flips, we might obtain:

70% heads

After 100 flips:

54% heads

After 1,000 flips:

50.8% heads

The exact results vary, but with many trials the experimental proportion often becomes closer to the theoretical probability.

This idea is extremely important in probability and statistics.

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Probability in Science

Scientists use probability when dealing with uncertainty.

Examples include:

• genetics
• radioactive decay
• weather prediction
• disease studies
• particle physics
• ecology
• experimental measurements

For example, genetics can predict the probability that offspring inherit particular combinations of alleles.

Probability does not identify exactly what will happen to every individual.

Instead, it describes the likelihood of possible outcomes.

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Probability and Risk

Probability is also used to describe risk.

Suppose a particular event has probability:

1 in 100

This can be written as:

1/100 = 0.01 = 1%

Risk information appears in areas such as:

• engineering
• insurance
• transportation
• medicine
• environmental science
• natural hazards

However, probability alone does not tell us how serious an event would be.

A low-probability event could still have very serious consequences.

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Probability in Everyday Decisions

We use probability thinking even when we do not calculate exact numbers.

For example:

"The train is usually on time, so it will probably arrive soon."

"Those clouds are very dark, so rain seems likely."

"Only one ticket out of 1,000 wins, so winning is unlikely."

These statements involve judgments about likelihood.

Mathematical probability allows us to express such ideas more precisely when sufficient information is available.

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Probability Language

Probability can be communicated using words.

Impossible

Probability = 0

Very unlikely

Probability close to 0

Unlikely

Probability less than 0.5

Even chance

Probability = 0.5

Likely

Probability greater than 0.5

Very likely

Probability close to 1

Certain

Probability = 1

The exact boundary between words such as "likely" and "very likely" depends on context, so numerical probabilities are often more precise.

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Comparing Probability Statements

Consider:

Event A: 15%

Event B: 40%

Event C: 50%

Event D: 85%

We can say:

• D is more likely than C
• C is more likely than B
• B is more likely than A
• C has an even chance
• A is unlikely
• D is very likely

But none except an event with probability 100% is mathematically certain.

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Worked Example: Marbles

A bag contains:

5 red marbles

3 blue marbles

2 green marbles

Total:

5 + 3 + 2 = 10 marbles

Therefore:

P(red) = 5/10 = 0.5

P(blue) = 3/10 = 0.3

P(green) = 2/10 = 0.2

We can compare:

red > blue > green

Red is the most likely colour to be selected.

Green is the least likely.

Selecting red has an even chance.

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Worked Example: A Die

For a fair six-sided die:

P(rolling 6)

= 1/6

≈ 16.7%

P(rolling greater than 3)

Possible favourable outcomes:

4, 5, 6

P(greater than 3) = 3/6

= 50%

P(rolling less than 7)

All six outcomes work.

P(less than 7) = 6/6

= 100%

P(rolling 8)

There is no favourable outcome.

P(8) = 0/6

= 0%

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Worked Example: Interpreting Real-World Probability

Suppose a weather forecast gives:

P(rain) = 0.25

Convert to percentage:

0.25 × 100 = 25%

This means rain is considered possible but relatively unlikely.

Probability of no rain:

1 − 0.25 = 0.75

= 75%

No rain is therefore more likely than rain.

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Common Mistakes

Thinking unlikely means impossible

A 5% probability is small, but the event can still happen.

Thinking likely means certain

A 90% probability still allows the event not to happen.

Writing probabilities greater than 1

For example:

1.4

cannot be a probability.

Writing negative probabilities

−0.2

cannot be a probability.

Assuming a fair coin must alternate

A fair coin can produce:

Heads, Heads, Heads, Heads

Probability does not require alternating outcomes.

Expecting exact results from a small number of trials

A probability of 50% does not guarantee exactly five successes in ten trials.

Confusing an outcome with an event

An outcome is one possible result.

An event may contain several outcomes.

Forgetting to count all possible outcomes

Always identify the full sample space when calculating simple theoretical probabilities.

Assuming all outcomes are equally likely

The simple formula:

favourable outcomes / total outcomes

requires outcomes to be equally likely.

Confusing probability with prediction

Probability describes likelihood. It does not tell us exactly what will happen next.

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Key Terms

Probability — A numerical measure of how likely an event is to occur.

Likelihood — How likely or unlikely something is to happen.

Probability scale — A scale from 0 to 1 representing likelihood.

Impossible event — An event with probability 0.

Certain event — An event with probability 1.

Likely event — An event with a relatively high probability of occurring.

Unlikely event — An event with a relatively low probability of occurring.

Even chance — A probability of 1/2, 0.5, or 50%.

Outcome — One possible result of an experiment or situation.

Event — One outcome or a collection of outcomes being considered.

Sample space — The complete set of possible outcomes.

Favourable outcome — An outcome that satisfies the event being investigated.

Equally likely — Having the same probability of occurring.

Fair — Describes a probability device in which intended equivalent outcomes have equal probabilities.

Random — Involving outcomes that cannot be predicted with certainty beforehand.

Trial — One performance of a probability experiment.

Theoretical probability — Probability calculated from known possible outcomes.

Experimental probability — Probability estimated from results observed during repeated trials.

Empirical probability — Another term for probability estimated from observed data.

Relative frequency — The proportion of trials in which an event occurs.

Complement — The event that the original event does not occur.

Complementary events — Events representing an event occurring and not occurring; their probabilities total 1.

Risk — The possibility of an unwanted event, often described partly using probability.

Uncertainty — Lack of complete certainty about an outcome.

Random variation — Natural differences in results produced by random processes.

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Key Takeaways

• Probability measures how likely an event is to occur.
• Probability ranges from 0 to 1.
• Probability can also be expressed from 0% to 100%.
• A probability of 0 represents an impossible event.
• A probability of 1 represents a certain event.
• A probability of 0.5 represents an even chance.
• Values closer to 0 represent less likely events.
• Values closer to 1 represent more likely events.
• Probability can be expressed as fractions, decimals, or percentages.
• For equally likely outcomes:

Probability = favourable outcomes / total possible outcomes

• A sample space contains all possible outcomes.
• An outcome is one possible result.
• An event can contain one or more outcomes.
• Probabilities allow events to be compared mathematically.
• An unlikely event is still possible.
• A likely event is not necessarily certain.
• Probability describes likelihood rather than guaranteeing what will happen next.
• Complementary probabilities total 1, or 100%.
• Theoretical probability is calculated from known possible outcomes.
• Experimental probability is estimated from observed results.
• Experimental probability may differ from theoretical probability, especially when few trials are conducted.
• With many repeated trials, experimental results often become closer to theoretical expectations.
• Probability is used in science, weather forecasting, sports, engineering, genetics, insurance, risk analysis, and everyday decisions.
• Probability should always be interpreted in the context of the event being described.

When interpreting probability, ask:

What event is being considered?

What are all the possible outcomes?

Are the outcomes equally likely?

Where would the event fall on the probability scale?

Is it impossible, unlikely, an even chance, likely, or certain?

What does the probability actually tell me—and what does it not tell me?

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Check Your Understanding

1. What does probability measure?

2. What is the smallest possible probability?

3. What is the largest possible probability?

4. Place these events in order from least likely to most likely:

0.8, 0.2, 0.5, 0.95, 0.05

5. Match each probability to the most appropriate description:

0
0.2
0.5
0.8
1

Descriptions:

certain, likely, impossible, unlikely, even chance

6. Convert 0.35 to a percentage.

7. Convert 75% to a decimal.

8. Convert 1/2 to a decimal and percentage.

9. A fair six-sided die is rolled. What is the probability of rolling a 3?

10. What is the probability of rolling an even number?

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

12. What is the probability of rolling a 9?

13. What is the probability of rolling a number less than 7?

14. A bag contains 6 red, 3 blue, and 1 green marble. Calculate P(red), P(blue), and P(green).

15. Which colour in Question 14 is most likely to be selected?

16. Explain why an event with probability 0.05 is not impossible.

17. Explain why an event with probability 0.95 is not certain.

18. If P(rain) = 0.3, calculate P(no rain).

19. A fair coin is flipped ten times and produces eight heads. Does this prove that the coin is unfair? Explain.

20. Challenge: A spinner has 10 equal sections. Four are red, three are blue, two are green, and one is yellow. Calculate the probability of each colour, express each as a fraction, decimal, and percentage, place the four events in order from least likely to most likely, and explain which events would be described as unlikely or having an even chance.