June 2025 Paper 3 Q15
15 The proportion of diners at a restaurant who are vegan is 20%
The number of diners at the restaurant who are vegan can be modelled by a binomial distribution.
(a) In a random sample of 25 diners:
(i) find the probability that no diner is a vegan. [1 mark]
(ii) find the probability that fewer than 6 diners are vegan. [1 mark]
(iii) find the probability that at least 9 diners are vegan. [2 marks]
(b) State one assumption necessary for the binomial distribution to be valid in the context of this question. [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains AWFW [0.00377, 0.0038] | B1 | 1.1b |
| (1) | ||
| (ii) Obtains AWFW [0.616, 0.62] | B1 | 1.1b |
| (1) | ||
| (iii) Finds \(\mathrm{P}(X \leqslant 8)\) = [0.953, 0.954] or \(\mathrm{P}(X \leqslant 9)\) = [0.982, 0.983] or \(\mathrm{P}(X \gt 9)\) = [0.017, 0.0174] or \(\mathrm{P}(X \geqslant 9)\) = [0.0467, 0.047] Condone incorrect or no labels | M1 | 1.1a |
| Obtains AWFW [0.0467, 0.047] | A1 | 1.1b |
| (2) |
Typical solution
(a)(i)
0.00378
(a)(ii)
0.617
(a)(iii)
\[\mathrm{P}(X \geqslant 9) = 1 - 0.9532\]\[= 0.0468\]| Scheme | Marks | AO |
|---|---|---|
States one of the following assumptions in context
Must use ‘vegan’ and ‘diner’ Allow equivalent statements for non-vegan diners for the reference to probability or independence Do not allow statements which imply probabilities are independent Ignore statement which implies there is a fixed number of diners | E1 | 3.5b |
| (1) | ||
| (5 marks) |
Typical solution
The event of a diner is vegan is independent from one diner to the next.