Probability basics

Probability measures how likely an event is, from 0 to 1. It can be a fraction, a decimal or a percentage.

When all the outcomes are equally likely:

P(event) = number of ways it can happen ÷ total number of outcomes

Example: a bag has 3 red, 5 blue and 2 green counters. One is taken at random. Find P(blue) and P(yellow).

1. 5 blue out of 3 + 5 + 2 = 10 counters
2. P(blue) = 5/10 = ½ (0.5 or 50%)
3. There are no yellow counters: P(yellow) = 0, impossible

The probabilities of all the possible outcomes add up to 1. So:

P(not A) = 1 − P(A) (learn this)

Events are mutually exclusive if they can't happen at the same time. Then P(A or B) = P(A) + P(B).

Example: a spinner lands on each colour with these probabilities. Find x and P(not red).

1. 0.3 + 0.25 + x + 0.15 = 1, so 0.7 + x = 1
2. x = 0.3
3. P(not red) = 1 − 0.3 = 0.7

List outcomes systematically, in an order, so none are missed or repeated. For two events, a sample space diagram (a grid) shows every combination.

Example: two spinners numbered 1 to 4 are spun and the scores added. Find P(total of 5).

1. There are 4 × 4 = 16 outcomes
2. A total of 5 appears 4 times
3. P(total of 5) = 4/16 = ¼

Example: list every order of the letters A, B and C.

1. Start with A: ABC, ACB. Then B: BAC, BCA. Then C: CAB, CBA
2. 6 orders

If one choice can be made in m ways and another in n ways, together they can be made in m × n ways. This works for any number of choices.

Example: a café offers 3 starters, 5 mains and 2 desserts. How many different three-course meals are there?

1. 3 × 5 × 2 = 30 meals

Example: a 4-digit code uses the digits 0 to 9. How many codes are there, and how many if no digit can be repeated?

1. Each digit has 10 choices: 10 × 10 × 10 × 10 = 10 000
2. Without repeats, there is one fewer choice each time: 10 × 9 × 8 × 7 = 5040