June 2025 Paper 2 Q15
15 The probability distribution of the random variable \(X\) is modelled as follows.
- \(\mathrm{P}(X = 1) = p\) where \(p\) is a constant.
- \(\mathrm{P}(X = x) = 2\mathrm{P}(X = x - 1)\) for \(x = 2, 3, 4\).
- \(\mathrm{P}(X = x) = 0\) for all other values of \(x\).
Three values, \(X_1\), \(X_2\) and \(X_3\), of \(X\) are chosen at random.
Determine \(\mathrm{P}(X_3 \gt X_1 + X_2)\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(p + 2p + 4p + 8p = 1 \left[\Rightarrow p = \dfrac{1}{15}\right]\) | M1 | 3.1a |
| \(\begin{array}{|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(X = x) & \frac{1}{15} & \frac{2}{15} & \frac{4}{15} & \frac{8}{15} \\ \hline\end{array}\) | A1 | 3.4 |
| \(\left(\frac{1}{15}\right)^2 \times \frac{4}{15} + \left(\frac{1}{15}\right)^2 \times \frac{8}{15} + \frac{1}{15} \times \frac{2}{15} \times \frac{8}{15} \times 2\) \(\left[= \left(\frac{1}{15}\right) \times \left(\frac{4}{15} + \frac{8}{15}\right) + \frac{1}{15} \times \frac{2}{15} \times \frac{8}{15} \times 2\right]\) | M1 | 3.4 |
| \(= \dfrac{44}{3375}\) | A1 | 1.1 |
| [4] |
Notes
M1: May be implied by first A1 (which is implied by use of 4 correct probabilities).
A1: Soi
M1: FT their probabilities, dep \(\Sigma p = 1\) Allow M1 if \(\times 2\) omitted or \(2 \times\) an additional term.
Allow this mark for \(44p^3\) May see e.g.
| \(X_1\) | \(X_2\) | \(X_1 + X_2\) | \(X_3\) | P |
|---|---|---|---|---|
| 1 | 1 | 2 | 3 | \(\left(\frac{1}{15}\right)^2 \times \frac{4}{15}\) |
| 1 | 1 | 2 | 4 | \(\left(\frac{1}{15}\right)^2 \times \frac{8}{15}\) |
| 1 | 2 | 3 | 4 | \(\frac{1}{15} \times \frac{2}{15} \times \frac{8}{15}\) |
| 2 | 1 | 3 | 4 | \(\frac{2}{15} \times \frac{1}{15} \times \frac{8}{15}\) |
Leading to an expression of the form given (3 or 4 terms each comprised of 3 of their probabilities multiplied together).
A1: Correct value from correct working implies all 4 marks (i.e. if candidates write \(44p^3\) and \(p = \frac{1}{15} \Rightarrow \mathrm{P}(\ldots) = \frac{44}{3375}\) then score 4/4)
Accept awrt 0.0130 (3sf)