# Probability: Bayes, counting and Bernoulli trials · Year 12

> Explanation, worked examples and free practice for the national curriculum in England.

In Probability: Bayes, counting and Bernoulli trials (Year 12) you practise 5 skills: The multiplication principle · Arrangements and roles · Choosing without order · Bernoulli trials · At least and at most: the complement.

In this topic probability is built from counting. When every outcome is equally likely, the probability is the number of outcomes that fit over the number of all outcomes. To count, use the multiplication principle: choose in steps, and multiply the numbers of options at each step. When the order matters, count arrangements; when it does not, divide by the number of orders of each group.

When the same trial is repeated several times and the trials are independent, the probability of a run of results is the product of the probabilities. The probability of exactly a given number of successes is the number of runs that fit times the probability of one run: these are Bernoulli trials. For at least once, it is easier to find the complement.

Bayes’ rule turns a question around. When a medical test comes out positive, the question is how likely it is that the person is really ill. A simple way is to think of a hundred thousand people, count each group in a table, and divide the ill people with a positive result by all the positive results. When the illness is rare, the answer is smaller than expected.

**Tip:** Before counting, ask whether the order matters. If it does, count arrangements; if not, divide by the number of orders.

## What you learn

### The multiplication principle

- Example: A restaurant has 2 starters, 5 main courses and 2 desserts. A meal is one starter, one main course and one dessert. How many different meals can be made?
- Answer: 20
- Solution: By the multiplication principle: 2 × 5 × 2 = 20 .

### Arrangements and roles

- Example: A class has 5 pupils. Four pupils are chosen for four different roles, one pupil for each role. In how many ways can this be done?
- Answer: 120
- Solution: The first role has 5 options, the second 4, and so on: 5 × 4 × 3 × 2 = 120 .

### Choosing without order

- Example: In a group of 4 friends, every two of them shake hands once. How many handshakes are there?
- Answer: 6
- Solution: Each handshake is a choice of two from 4, with no order: 4 × 3 ÷ 2 = 6 .

### Bernoulli trials

- Example: A spinner has 4 equal parts, and one of them is red. It is spun twice. What is the probability that it never lands on red?
- Answer: 9/16
- Solution: On each spin the probability of not landing on red is 3/4 . So the probability is (3/4)² , that is 9/16 .

### At least and at most: the complement

- Example: A spinner has 6 equal parts, and one of them is red. It is spun twice. What is the probability that it lands on red at least once?
- Answer: 11/36
- Solution: The complement is that it never lands on red: (5/6)² = 25/36 . So the probability is 1 − 25/36 , that is 11/36 .

The exercises are made by rules and the answers are worked out in code. The topic follows selected AS and A level mathematics practice in England (the GCE subject content).

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