Statistics

Probability

The probability scale, single and combined events, expected frequency, tree diagrams and without-replacement problems

Explanation & Worked Examples

The Probability Scale

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal or percentage.

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

For a fair six-sided die, P(rolling a 4) = 1/6 because there is 1 favourable outcome out of 6 equally likely ones.

The Complement

P(event does not happen) = 1 − P(event happens)

If P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7. The probabilities of all possible outcomes always add up to 1.

Sanity check: a probability can never be more than 1 or less than 0. If you ever get an answer like 1.4 or −0.2, something has gone wrong.

Combined Events

The AND rule (independent events) — multiply

If two events don't affect each other, the chance of both happening is the product of their probabilities:

P(A and B) = P(A) × P(B)

e.g. two coins: P(two heads) = ½ × ½ = ¼.

The OR rule (mutually exclusive events) — add

If two events can't both happen at once, the chance of one or the other is the sum:

P(A or B) = P(A) + P(B)

Expected frequency

Expected number = probability × number of trials

e.g. rolling a die 60 times, expected number of sixes = 1/6 × 60 = 10.

AND vs OR: "and" means both must happen → multiply (the answer gets smaller). "or" means either will do → add (the answer gets bigger).

Easy Examples

Single events — count the favourable outcomes over the total.

Example 1 — A Fair Die

What is the probability of rolling an even number on a fair six-sided die?

1Even numbers are 2, 4, 6 → 3 favourable outcomes out of 6
2P(even) = 3/6 = 1/2
Example 2 — Coloured Counters

A bag has 3 red and 2 blue counters. One is taken at random. What is P(red)?

1Total counters = 3 + 2 = 5
2P(red) = 3/5
Example 3 — The Complement

Using the same bag, what is P(not red)?

1P(not red) = 1 − P(red) = 1 − 3/5
2= 2/5 (the 2 blue counters)

Medium Examples

Expected frequency, the complement, and mutually exclusive events.

Example 1 — Expected Frequency

A fair die is rolled 60 times. How many sixes would you expect?

1Expected = P(six) × trials = 1/6 × 60
2= 10 sixes
Example 2 — Mutually Exclusive (OR rule)

A spinner lands on red with probability 0.2 and blue with probability 0.3. What is P(red or blue)?

1They can't both happen, so add: 0.2 + 0.3
2= 0.5
Example 3 — Find a Missing Probability

A spinner can land on red, blue or green. P(red) = 0.2 and P(blue) = 0.3. Find P(green).

1All probabilities add to 1: P(green) = 1 − (0.2 + 0.3)
2= 1 − 0.5 = 0.5
Tip: expected frequency is rarely a whole number in real trials — but the expected value (probability × trials) is what the question wants.

Complex Examples

Independent events (AND), and picking without replacement.

Example 1 — Two Independent Events

A coin is tossed and a die is rolled. What is P(heads and a six)?

1Independent, so multiply: P(heads) × P(six) = 1/2 × 1/6
2= 1/12
Example 2 — Without Replacement

A bag has 5 red and 3 blue counters. Two are taken without replacement. What is P(both red)?

1First red: 5/8. Now 4 red of 7 remain, so second red: 4/7
2P(both red) = 5/8 × 4/7 = 20/56 = 5/14
Example 3 — At Least One

A coin is tossed three times. What is P(at least one head)?

1Use the complement: P(at least one head) = 1 − P(no heads)
2P(all tails) = 1/2 × 1/2 × 1/2 = 1/8
3P(at least one head) = 1 − 1/8 = 7/8
"Without replacement": after the first pick the totals change — both the numerator (favourable left) and the denominator (total left) drop by one. "At least one" problems are almost always quickest via 1 − P(none).
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