June 2023 Paper 3 Q13
13 There are two types of coins in a money box:
- 20% are bronze coins
- 80% are silver coins
Craig takes out a coin at random and places it back in the money box.
Craig then takes out a second coin at random.
(a) Find the probability that both coins were of the same type. [2 marks]
(b) Find the probability that both coins are bronze, given that at least one of the coins is bronze. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Finds P(both bronze) or P(both silver) or calculates \(1 - 2 \times 0.2 \times 0.8\) PI by correct answer | M1 | 3.1b |
| Obtains the correct probability Ignore incorrect rounding after correct probability seen | A1 | 1.1b |
| (2) |
Typical solution
P(both bronze) = \(0.2 \times 0.2 = 0.04\)
P(both silver) = \(0.8 \times 0.8 = 0.64\)
P(both same type) = 0.68
| Scheme | Marks | AO |
|---|---|---|
| Finds P(at least one of the coins is bronze) | M1 | 3.1b |
| Obtains the correct probability Allow 0.11 or better for \(\dfrac{1}{9}\) Ignore incorrect rounding after correct probability seen | A1 | 2.2a |
| (2) | ||
| (4 marks) |
Typical solution
P(at least one bronze)
\[= 1 - 0.8 \times 0.8\]\[= 0.36\]P(both bronze | at least one bronze)
\[= \frac{0.2 \times 0.2}{0.36} = \frac{1}{9}\]