Probability Concepts

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

Probability is used whenever an outcome is uncertain.

Examples include:

• tossing a coin
• rolling a die
• choosing a card
• predicting tomorrow's weather
• estimating the chance of a machine failing
• predicting whether a sports team will win

Probability allows us to describe uncertainty using numbers.

A probability is always between 0 and 1.

0 ≤ P(event) ≤ 1

What Is Probability?

The probability of an event describes how likely that event is to occur.

Probability can be written as:

• a fraction
• a decimal
• a percentage

For example:

P(event) = 1/2

This can also be written as:

P(event) = 0.5

or:

P(event) = 50%

These all represent the same probability.

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The Probability Scale

Probability values range from 0 to 1.

A probability of:

0 means the event is impossible.

1 means the event is certain.

A probability close to 0 means the event is unlikely.

A probability close to 1 means the event is likely.

A probability of 0.5 means the event is equally likely to happen or not happen.

For example:

P(rolling a 7 on a normal six-sided die) = 0

P(rolling a number from 1 to 6) = 1

P(getting heads on a fair coin) = 0.5

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What Is a Random Event?

A random event is an event whose individual outcome cannot be predicted with certainty beforehand.

For example, when tossing a fair coin, we know the possible outcomes are:

• heads
• tails

However, we cannot know for certain which one will occur on a particular toss.

Similarly, when rolling a die, the possible outcomes are:

1, 2, 3, 4, 5, 6

The exact result of one roll is random.

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Outcomes and Events

An outcome is one possible result of a random experiment.

An event is one or more outcomes that we are interested in.

Example:

A die is rolled.

The possible outcomes are:

1, 2, 3, 4, 5, 6

The event "roll an even number" contains:

2, 4, 6

The event "roll a number greater than 4" contains:

5, 6

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Sample Space

The sample space is the set of all possible outcomes.

For a coin toss:

Sample space = {H, T}

For a six-sided die:

Sample space = {1, 2, 3, 4, 5, 6}

For a spinner with four equal sections labeled A, B, C, D:

Sample space = {A, B, C, D}

Writing the sample space can help make probability calculations easier.

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Calculating Simple Probability

When all outcomes are equally likely:

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

In symbols:

P(event) = favourable outcomes ÷ total outcomes

For example, suppose a fair die is rolled.

What is the probability of rolling a 4?

There is 1 favourable outcome.

There are 6 possible outcomes.

Therefore:

P(4) = 1/6

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Example: Rolling an Even Number

A fair six-sided die is rolled.

What is the probability of rolling an even number?

Possible outcomes:

1, 2, 3, 4, 5, 6

Even outcomes:

2, 4, 6

Number of favourable outcomes = 3

Total outcomes = 6

Therefore:

P(even) = 3/6

Simplify:

P(even) = 1/2

So the probability is:

1/2 = 0.5 = 50%

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Example: Rolling a Number Greater Than 4

Possible outcomes:

1, 2, 3, 4, 5, 6

Numbers greater than 4:

5, 6

Therefore:

P(number greater than 4) = 2/6

Simplify:

P(number greater than 4) = 1/3

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Probability with Coins

A fair coin has two equally likely outcomes:

• heads
• tails

Therefore:

P(heads) = 1/2

P(tails) = 1/2

If the coin is fair, neither side is more likely.

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Probability with Spinners

Suppose a spinner is divided into 8 equal sections.

Three sections are red.

What is the probability of landing on red?

Number of red sections = 3

Total sections = 8

Therefore:

P(red) = 3/8

This calculation works because all eight sections are equal in size and therefore equally likely.

If sections have different sizes, simply counting sections may not give the correct probability.

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Theoretical Probability

Theoretical probability is the probability calculated from the possible outcomes of an experiment.

It is based on what we expect to happen mathematically.

For equally likely outcomes:

Theoretical probability = favourable outcomes ÷ total possible outcomes

Example:

A fair die has six sides.

Two sides show numbers greater than 4:

5 and 6.

Therefore:

P(number greater than 4) = 2/6 = 1/3

This is the theoretical probability.

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Experimental Probability

Experimental probability is based on what actually happens when an experiment is performed.

It is sometimes called relative frequency.

The formula is:

Experimental probability = number of times event occurs ÷ total number of trials

For example, a coin is tossed 50 times.

Heads occurs 28 times.

Therefore:

Experimental P(heads) = 28/50

Experimental P(heads) = 0.56

or:

56%

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Theoretical vs Experimental Probability

Theoretical probability is based on mathematical reasoning.

Experimental probability is based on observed results.

The two probabilities may not be exactly equal.

This is normal.

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Why Experimental Probability Can Differ

Suppose a fair coin is tossed 10 times.

It might produce:

6 heads and 4 tails.

Experimental P(heads) = 6/10 = 0.6

But the theoretical probability is:

P(heads) = 0.5

The difference occurs because random variation affects small samples.

Theoretical probability tells us what we expect in the long run, not what must happen in every short experiment.

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The Law of Large Numbers

When the number of trials becomes very large, experimental probability tends to move closer to theoretical probability.

For example:

10 coin tosses might produce 70% heads.

100 tosses might produce 54% heads.

10,000 tosses might produce something much closer to 50%.

This idea is known as the law of large numbers.

The key idea is:

more trials → experimental probability usually becomes more stable

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Complementary Events

Two events are complementary when one event happens exactly when the other does not.

Examples include:

• heads and not heads
• rain and no rain
• winning and not winning
• rolling an even number and rolling an odd number

For complementary events:

P(event) + P(not event) = 1

Therefore:

P(not event) = 1 - P(event)

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Example: Complementary Probability

Suppose:

P(rain) = 0.3

Then:

P(no rain) = 1 - 0.3

P(no rain) = 0.7

Therefore, the probability of no rain is:

0.7 or 70%

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Example: Rolling a Die

What is the probability of not rolling a 6?

P(6) = 1/6

Therefore:

P(not 6) = 1 - 1/6

P(not 6) = 5/6

We can also calculate this by counting the outcomes:

1, 2, 3, 4, 5

There are 5 favourable outcomes out of 6.

So:

P(not 6) = 5/6

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Complementary Events on a Probability Scale

Suppose an event has probability:

P(A) = 0.72

Its complement has probability:

P(not A) = 1 - 0.72

P(not A) = 0.28

Together:

0.72 + 0.28 = 1

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Impossible and Certain Events

Some events are impossible.

For example:

Rolling an 8 on a normal six-sided die.

P(rolling 8) = 0

Some events are certain.

For example:

Rolling a number less than 7 on a standard six-sided die.

P(number less than 7) = 1

These values form the two ends of the probability scale.

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Equally Likely Outcomes

An important assumption in many simple probability calculations is that outcomes are equally likely.

For example, a fair die has six equally likely outcomes.

Therefore each result has probability:

1/6

However, not every situation has equally likely outcomes.

A spinner with unequal sections may make some outcomes more likely than others.

A biased coin may not have a 50% chance of heads.

Always consider whether the outcomes really are equally likely.

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Interpreting Probability in Context

Probability does not tell us exactly what will happen.

It tells us how likely something is.

Suppose a weather forecast says:

P(rain) = 0.8

This means there is an 80% probability of rain under the conditions represented by the forecast.

It does not mean:

• it will rain for 80% of the day
• 80% of the city must receive rain
• rain is guaranteed

Probability describes uncertainty, not certainty.

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Interpreting a Small Probability

Suppose:

P(machine failure today) = 0.01

This means there is a:

1% probability of failure

and a:

99% probability of no failure

A 1% probability is small, but the event is not impossible.

This distinction is important.

Unlikely does not mean impossible.

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Interpreting a Large Probability

Suppose:

P(team wins) = 0.85

The team is very likely to win.

However, there is still:

P(team does not win) = 1 - 0.85

P(team does not win) = 0.15

So an unexpected result is still possible.

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Fractions, Decimals, and Percentages

Probabilities can be expressed in several forms.

Being able to move between these forms helps when interpreting probabilities.

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Converting Between Forms

To convert a fraction to a decimal:

divide numerator by denominator

Example:

3/5 = 3 ÷ 5 = 0.6

To convert a decimal to a percentage:

multiply by 100

0.6 × 100 = 60%

Therefore:

3/5 = 0.6 = 60%

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Probability from a Table

Suppose a bag contains:

Total objects:

4 + 3 + 2 + 1 = 10

Therefore:

P(red) = 4/10 = 2/5

P(blue) = 3/10

P(green) = 2/10 = 1/5

P(yellow) = 1/10

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Worked Example: Simple Probability

A bag contains:

• 5 red counters
• 3 blue counters
• 2 green counters

One counter is chosen at random.

What is the probability of choosing blue?

Total counters:

5 + 3 + 2 = 10

Blue counters = 3

Therefore:

P(blue) = 3/10

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Worked Example: Complement

Using the same bag:

What is the probability of not choosing blue?

Method 1:

P(not blue) = 1 - 3/10

P(not blue) = 7/10

Method 2:

Not blue counters:

5 red + 2 green = 7

Therefore:

P(not blue) = 7/10

Both methods give the same result.

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Worked Example: Experimental Probability

A basketball player takes 80 free throws and scores 52.

Estimate the probability that the player scores the next free throw based on the experiment.

Experimental probability:

P(score) = 52/80

P(score) = 0.65

Estimated probability:

0.65 or 65%

This does not guarantee the next shot will be successful.

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Worked Example: Theoretical vs Experimental

A fair die is rolled 120 times.

A 6 appears 24 times.

Theoretical probability:

P(6) = 1/6

P(6) ≈ 0.167

Experimental probability:

P(6) = 24/120

P(6) = 0.20

The values are different because experimental results are affected by random variation.

With more trials, the experimental probability would generally be expected to become closer to the theoretical value.

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Worked Example: Interpreting Probability

A medicine has a probability of 0.92 of producing a particular desired response in a population studied.

What does 0.92 mean?

It means the event has an estimated probability of:

92%

It does not mean the outcome is guaranteed for an individual.

There is still:

1 - 0.92 = 0.08

or:

8%

probability of the complementary outcome under the model being used.

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Common Misconceptions

If a coin lands heads five times, tails must come next.

Incorrect. If each toss is independent and the coin is fair, the probability of tails on the next toss is still 1/2.

A probability of 0.9 means the event will definitely happen.

Incorrect. It means the event is very likely, not certain.

Experimental probability must equal theoretical probability.

Incorrect. Experimental results can vary because of randomness.

A probability of 0 means unlikely.

A probability of 0 means impossible in the model.

A probability of 1 means very likely.

A probability of 1 means certain.

If an event has probability 0.4, its complement also has probability 0.4.

Incorrect.

P(complement) = 1 - 0.4 = 0.6.

Ten coin tosses should always produce exactly five heads and five tails.

Incorrect. 50% is a long-run expectation, not a guarantee for a small number of trials.

Did You Know?

Casinos, insurance companies, weather forecasters, scientists, engineers, doctors, and financial analysts all use probability.

They often cannot predict exactly what will happen in one individual case.

Instead, probability allows them to estimate how frequently different outcomes are expected to occur across many cases.

That is one of the most important ideas in probability:

probability describes patterns in uncertainty rather than guaranteeing individual outcomes.

Key Terms

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

Random event – An event whose individual outcome cannot be predicted with certainty beforehand.

Outcome – One possible result of a random experiment.

Event – One or more outcomes of interest.

Sample space – The set of all possible outcomes.

Theoretical probability – Probability calculated from the possible outcomes.

Experimental probability – Probability estimated from observed results.

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

Complementary events – Two events in which one occurs exactly when the other does not.

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

Certain event – An event with probability 1.

Impossible event – An event with probability 0.

Key Takeaways

• Probability measures how likely an event is to happen.
• Probabilities range from 0 to 1.
• A probability of 0 means impossible.
• A probability of 1 means certain.
• A random event has an uncertain individual outcome.
• For equally likely outcomes, probability is calculated using favourable outcomes ÷ total possible outcomes.
• Theoretical probability is calculated mathematically.
• Experimental probability is based on observed results.
• Experimental and theoretical probability may differ, especially with a small number of trials.
• As the number of trials increases, experimental probabilities often become more stable and approach theoretical probabilities.
• Complementary probabilities add to 1.
• P(not A) = 1 - P(A).
• Probability can be expressed as fractions, decimals, or percentages.
• A high probability does not guarantee an event.
• A low probability does not make an event impossible.
• Probabilities should always be interpreted in the context of the situation.