Probability

Probability measures how likely an event is to occur. It is expressed as a value between 0 (impossible) and 1 (certain), as a fraction, decimal or percentage.

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The Probability Scale

On the scale from 0 to 1: 0 = impossible, 0.5 = equally likely (evens), 1 = certain.

P(event) denotes the probability of an event occurring. P(impossible event) = 0; P(certain event) = 1; P(fair coin shows heads) = 1/2

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Theoretical Probability

When outcomes are equally likely:

P(event) = number of favourable outcomes ÷ total number of outcomes

Example: A fair six-sided die is rolled. P(prime number) = ? Prime numbers on a die: 2, 3, 5 → 3 favourable outcomes out of 6. P(prime) = 3/6 = 1/2

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Complementary Events

P(event does NOT happen) = 1 − P(event)

Example: P(rain tomorrow) = 0.3 → P(no rain) = 1 − 0.3 = 0.7

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Mutually Exclusive Events

Events are mutually exclusive if they cannot both happen at the same time.

For mutually exclusive events A and B: P(A or B) = P(A) + P(B)

Example: Rolling a die: P(2 or 5) = 1/6 + 1/6 = 1/3

For an exhaustive set of mutually exclusive events, all probabilities sum to 1.

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Experimental Probability (Relative Frequency)

When outcomes are not equally likely, use experiments.

P(event) ≈ number of times event occurs ÷ total number of trials

Example: A biased coin is flipped 200 times. Heads appears 130 times. Experimental P(heads) = 130/200 = 0.65

Key principle: As the number of trials increases, experimental probability gets closer to the true theoretical probability.

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Expected Frequency

Expected frequency = probability × number of trials

Example: A fair die is rolled 90 times. How many times would you expect a 6? Expected = (1/6) × 90 = 15 times

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Sample Spaces

A sample space lists all possible outcomes. For combined experiments, use a two-way table.

Example: Rolling two dice — the table below shows all possible totals (rows = die 1, columns = die 2):

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

P(total = 8) = 5/36 (five ways: 2+6, 3+5, 4+4, 5+3, 6+2)

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Tree Diagrams

A tree diagram shows outcomes of successive events. Multiply probabilities along branches; add probabilities for separate branches.

Example: A bag has 3 red and 5 blue counters. A counter is drawn, replaced, then a second is drawn.

• P(Red, Red) = (3/8) × (3/8) = 9/64
• P(Red, Blue) = (3/8) × (5/8) = 15/64
• P(Blue, Red) = (5/8) × (3/8) = 15/64
• P(Blue, Blue) = (5/8) × (5/8) = 25/64