| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Distribution |
| Type | At least one success |
| Difficulty | Moderate -0.8 This is a straightforward application of binomial distribution formulas with no conceptual challenges. Part (a) requires direct calculation using B(n,p) with given parameters, and part (b) involves simple algebraic manipulation of expectation (np ≥ 6) and complement probability (1 - 0.8^n > 0.95). All techniques are standard S2 bookwork with no problem-solving insight required. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(P(X=2) = {}^{14}C_2 \times 0.2^2 \times 0.8^{12}\) | M1 | For writing or using \({}^{14}C_2 \times 0.2^2 \times 0.8^{12}\) (Allow 91 for \({}^{14}C_2\)) |
| \(= 0.2501\) | A1 | awrt 0.2501. NB 0.2501 with no working scores M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(P(X>3) = 1 - P(X \leqslant 3) = 1 - 0.2340\) | M1 | For writing or using \(1 - P(X \leqslant 3)\) |
| \(= 0.7660\) | A1 | awrt 0.766. NB awrt 0.766 with no working scores M1A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(np = 6 \Rightarrow n = \frac{6}{0.2}\) | M1 | For use of \(np = 6\), e.g. \(0.2n = 6\) (Allow \(\geqslant\)) |
| \(n = 30\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(1 - P(Y=0) > 0.95 \Rightarrow P(Y=0) < 0.05\) | M1 | For writing or using \(P(Y \geqslant 1) = 1 - P(Y=0)\) (Allow \(P(Y \geqslant 1) = 1 - P(Y \leqslant 0)\)) |
| \(0.8^n < 0.05\) | M1 | For \(0.8^n < 0.05\) oe (Allow \(=\) or \(\leqslant\)) |
| \(0.8^{14} = 0.04398...\ [< 0.05]\) or \(n > \frac{\ln 0.05}{\ln 0.8} \Rightarrow n > 13.425\) | dM1 | Dependent on previous M1. For substitution of \(n\) (allow \(0.8^{13} = 0.05497...\)) or rearranging to \(n > ...\) (Allow \(=\) or \(\geqslant\)). If using logs allow any base e.g. \(n > \log_{0.8} 0.05\) |
| \(n = 14\) | A1 | cao |
## Question 1:
### Part (a)(i)
$X \sim B(14, 0.2)$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $P(X=2) = {}^{14}C_2 \times 0.2^2 \times 0.8^{12}$ | M1 | For writing or using ${}^{14}C_2 \times 0.2^2 \times 0.8^{12}$ (Allow 91 for ${}^{14}C_2$) |
| $= 0.2501$ | A1 | awrt 0.2501. **NB** 0.2501 with no working scores M1A1 |
### Part (a)(ii)
$X \sim B(25, 0.2)$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $P(X>3) = 1 - P(X \leqslant 3) = 1 - 0.2340$ | M1 | For writing or using $1 - P(X \leqslant 3)$ |
| $= 0.7660$ | A1 | awrt 0.766. **NB** awrt 0.766 with no working scores M1A1 |
### Part (b)(i)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $np = 6 \Rightarrow n = \frac{6}{0.2}$ | M1 | For use of $np = 6$, e.g. $0.2n = 6$ (Allow $\geqslant$) |
| $n = 30$ | A1 | cao |
### Part (b)(ii)
$Y \sim B(n, 0.2)$, require $P(Y \geqslant 1) > 0.95$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $1 - P(Y=0) > 0.95 \Rightarrow P(Y=0) < 0.05$ | M1 | For writing or using $P(Y \geqslant 1) = 1 - P(Y=0)$ (Allow $P(Y \geqslant 1) = 1 - P(Y \leqslant 0)$) |
| $0.8^n < 0.05$ | M1 | For $0.8^n < 0.05$ oe (Allow $=$ or $\leqslant$) |
| $0.8^{14} = 0.04398...\ [< 0.05]$ or $n > \frac{\ln 0.05}{\ln 0.8} \Rightarrow n > 13.425$ | dM1 | Dependent on previous M1. For substitution of $n$ (allow $0.8^{13} = 0.05497...$) or rearranging to $n > ...$ (Allow $=$ or $\geqslant$). If using logs allow any base e.g. $n > \log_{0.8} 0.05$ |
| $n = 14$ | A1 | cao |\begin{enumerate}
\item Sam is a telephone sales representative.
\end{enumerate}
For each call to a customer
\begin{itemize}
\item Sam either makes a sale or does not make a sale
\item sales are made independently
\end{itemize}
Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2\\
(a) Find the probability that Sam makes\\
(i) exactly 2 sales in 14 calls,\\
(ii) more than 3 sales in 25 calls.
Sam makes $n$ calls each day.\\
(b) Find the minimum value of $n$\\
(i) so that the expected number of sales each day is at least 6\\
(ii) so that the probability of at least 1 sale in a randomly selected day exceeds 0.95
\hfill \mbox{\textit{Edexcel S2 2023 Q1 [10]}}