# Independent and dependent events · Year 10

> Independent and dependent events, Year 10: Dependent or independent and more. Explanation, worked examples and free practice for the national curriculum in England.

In Independent and dependent events (Year 10) you practise 5 skills: Dependent or independent · Independent events together · Two dependent events · At least and at most · Which option to choose.

Two events are independent when the result of one does not change the chance of the other: rain on days whose weather is independent of each other, or two balls from a bag when the first is put back. When the first ball is not put back, one ball fewer is left, and the chance at the second draw depends on the first: the events are dependent.

When events are independent, the probability that all of them happen is the product of their probabilities, even for three events. The probability that an event does not happen is one minus the probability that it does, so a case where some events happen and some do not can be worked out too.

For two dependent events, build a tree of two stages. At the second stage use what is left after the first, multiply along each path, and add the paths that belong to the event.

The event “at least one” is easiest through the complement: one minus the probability that none of the events happens. The event “at most two” of three is one minus the probability that all three happen. To choose between options, work out the chance of each and compare.

**Tip:** Before working anything out, ask: does the first result change what can happen next?

## What you learn

### Dependent or independent

- Example: A bag holds 3 green balls and 6 white balls. A ball is drawn, its colour is noted, and it is put back in the bag. Then a second ball is drawn. Are the events “the first ball is green” and “the second ball is green” dependent or independent?
- Options: Dependent · Independent
- Answer: Independent
- Solution: The first ball goes back, so at the second draw there are again 9 balls, 3 of them green, whatever the first draw gave. So the events are independent.

### Independent events together

- Example: A bag holds 10 balls, and 3 of them are blue. A ball is drawn, its colour is noted, and it is put back in the bag. Then a second ball is drawn. What is the probability that both balls are blue? Write the answer as a fraction in its simplest form.
- Answer: 9/100
- Solution: The ball is put back, so at each draw the probability of blue is 3/10, and the draws are independent. Multiply: 3/10 × 3/10 = 9/100

### Two dependent events

- Example: A scratch card has 6 squares, and a prize is hidden in 2 of them. One square is scratched, then a second one. What is the probability that both squares show a prize? Write the answer as a fraction in its simplest form.
- Answer: 1/15
- Solution: At the first stage the probability is 2/6. Then 5 are left and only one of them fits, so at the second stage it is 1/5. Multiply: 2/6 × 1/5 = 2/30 = 1/15

### At least and at most

- Example: A wheel is split into 10 equal parts, and 2 of them are red. The wheel is spun twice. What is the probability that it stops on red at least once? Write the answer as a fraction in its simplest form.
- Answer: 9/25
- Solution: The complement: it never stops on red. On each spin that has probability 8/10, and the spins are independent. So: 8/10 × 8/10 = 64/100 The probability that it stops on red at least once is one minus this: 1 − 64/100 = 36/100 = 9/25

### Which option to choose

- Example: In one café, each customer gets a scratch card with 6 squares, and a cake is hidden in 4 of them. One square is scratched, then a second one, and the card wins if both show a cake. In a second café, each customer gets a scratch card with 7 squares, and a cake is hidden in 3 of them. One square is scratched, then a second one, and the card wins if both show a cake. In which café is the chance of winning larger?
- Options: In the second café · In the first café
- Answer: In the first café
- Solution: In the first café: 4/6 × 3/5 = 12/30 = 2/5 In the second café: 3/7 × 2/6 = 6/42 = 1/7 Compare the chances: 2/5 > 1/7 So the chance is larger in the first café.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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