June 2025 Paper 2 Q10
10
(a) The random variable \(W\) has the distribution \(\mathrm{B}\left(12, \frac{1}{3}\right)\).
Find \(\mathrm{P}(W \geqslant 7)\). [2]
Find \(\mathrm{P}(W \geqslant 7)\). [2]
(b) The random variable \(X\) has the distribution \(\mathrm{N}\left(45, 4^2\right)\).
Find \(\mathrm{P}(X \gt 50)\). [1]
Find \(\mathrm{P}(X \gt 50)\). [1]
(c) The random variable \(Y\) has the distribution \(\mathrm{N}\left(25, \sigma^2\right)\).
Given that \(\mathrm{P}(Y \lt 30) = 0.75\), find \(\sigma\). [2]
Given that \(\mathrm{P}(Y \lt 30) = 0.75\), find \(\sigma\). [2]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{P}(W \geqslant 7) = 1 - \mathrm{P}(W \leqslant 6)\) \([= 1 - 0.93355\ldots]\) | M1 | 1.1 |
| \(= 0.0664\) (3sf) | A1 | 1.1 |
| [2] |
Notes
M1: May be implied by A1 or 0.066 or 0.067 (2sf).
Allow \(1 - \mathrm{P}(W \leqslant 7)\,[= 1 - 0.9812\ldots]\) or 0.0188 for M1
A1: Accept awrt 0.0664 or awrt 0.0665 \([0.066447\ldots]\)
| Scheme | Marks | AO |
|---|---|---|
| 0.106 (3sf) | B1 | 1.1 |
| [1] |
Notes
B1: awrt 0.106
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{30 - 25}{\sigma} = \Phi^{-1}(0.75)\) | M1 | 3.4 |
| \(\sigma = \left[\dfrac{5}{0.67449} =\right] 7.41\) (3sf) | A1 | 1.1 |
| [2] |
Notes
M1: Attempt to standardise and equate to \(\Phi^{-1}(0.75)\,[= 0.67449]\)
May be implied by A1.
A1: awrt 7.41 or 7.42