Higher June 2021 Paper 2 Q13
13 Emilie takes part in two races.
The probability that she wins the first race is 0.7
The probability that she wins the second race is 0.4
The outcomes of the two races are independent.
(a) Complete the probability tree diagram. (2)
(b) Work out the probability that Emilie wins exactly one of the two races. (3)
Emilie is going to take part in a third race.
If she wins both of the first two races, the probability that she will win the third race is 0.6
If she wins exactly one of the first two races, the probability that she will win the third race is 0.3
(c) Work out the probability that Emilie will win exactly two of the three races. (3)
| Scheme | Marks |
|---|---|
| 0.3 | B1 |
| 0.6, 0.4, 0.6 | B1 |
| (2) |
Notes
B1: oe first race branch correct
B1: oe second race branches correct
| Scheme | Marks |
|---|---|
| \(0.7 \times \text{``}{0.6}\text{''}\ (= 0.42)\) oe or \(\text{``}{0.3}\text{''} \times \text{``}{0.4}\text{''}\ (= 0.12)\) oe or \(0.7 \times 0.4\ (= 0.28)\) oe or \(\text{``}{0.3}\text{''} \times \text{``}{0.6}\text{''}\ (= 0.18)\) oe | M1 |
| \(\text{``}{0.42}\text{''} + \text{``}{0.12}\text{''}\) oe or \(1 - \text{``}{0.28}\text{''} - \text{``}{0.18}\text{''}\) oe | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 0.54 | A1 |
| (3) |
Notes
M1: ft their tree diagram dep on probabilities being less than 1
M1: ft complete method to find probability that Emilie wins exactly one of the races
A1: oe, eg \(\dfrac{27}{50}\)
ft from their tree diagram on M marks only
| Scheme | Marks |
|---|---|
| \(0.7 \times 0.4 \times (1 - 0.6)\ (= 0.112)\) oe or \(\text{``}{0.54}\text{''} \times 0.3\ (= 0.162)\) oe or \(0.7 \times \text{``}{0.6}\text{''} \times 0.3 + \text{``}{0.3}\text{''} \times \text{``}{0.4}\text{''} \times 0.3\ (= 0.162)\) | M1 |
| eg \(\text{``}{0.112}\text{''} + \text{``}{0.162}\text{''}\) | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) NB: allow decimals, fractions or percentages with % as oe for probability Answer: 0.274 | A1 |
| (3) | |
| (8 marks) |
Notes
M1: ft
M1: ft For a fully correct method
A1: oe, eg \(\dfrac{137}{500}\)
ft from (a) and (b) on M marks only