Key Idea: Probability measures how likely an event is, from 0 (impossible) to 1 (certain). In 4.5 you count outcomes in a sample space, use "not A", read probabilities from tables and data, and find an expected number. Topic 4.6 then combines two or more events.
Core rules
| Rule | Formula | When to use |
|---|---|---|
| Basic probability | P(A) = n(A) ÷ n(U) | Equally likely outcomes: count A, divide by all. |
| Complement | P(A′) = 1 − P(A) | "Not A". |
| From a table | how many you want ÷ the total | A cell, a row or a column of a two-way table. |
| Experimental probability | frequency ÷ number of trials | Results of an experiment or survey. |
| Expected number | number of trials × P | How many times you expect it. It need not be whole. |
Key Idea: 12 of 30 students walk to school: P(walk) = 12/30 = 0.4, so P(not walk) = 0.6. Over 190 school days, a student who walks with probability 0.4 is expected to walk 0.4 × 190 = 76 days.
Use n(A) ÷ n(U) only when the outcomes are equally likely. Two dice: count the 36 pairs, not the 11 sums.
Show the fraction, e.g. 66/200, before the decimal: it earns the method mark if a later number slips. Give non-exact answers to 3 significant figures.
IB-style question [5 marks]
A café records the drinks bought by 200 customers.
| Coffee | Tea | |
|---|---|---|
| Under 25 | 46 | 24 |
| 25 or over | 88 | 42 |
One of these customers is chosen at random. (a) Find the probability that the customer bought tea.
Solution:
Add the tea column, then divide by the total in the question.
Final answer:
66/200 = 0.33
Part (b)
(b) Find the probability that the customer did not buy tea.
Solution:
"Not tea" is the complement: subtract from 1.
Final answer:
0.67
Part (c)
(c) The café expects 500 customers tomorrow. Estimate how many will buy tea.
Solution:
Expected number = number of customers × probability.
Final answer:
165 customers