| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Approximating the Binomial to the Poisson distribution |
| Type | Calculate single probability using Poisson approximation |
| Difficulty | Standard +0.3 This is a straightforward application of standard S2 techniques: part (a) uses binomial distribution with complement rule, part (b) is direct Poisson approximation (n large, p small), part (c) is basic uniform distribution probability, and part (d) combines Poisson probability with time scaling. All parts follow textbook methods with no novel insight required, though the multi-part structure and part (d)'s combination of concepts makes it slightly above average difficulty. |
| Spec | 2.04b Binomial distribution: as model B(n,p)2.04c Calculate binomial probabilities2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(D \sim B(8, 0.05)\) | M1 | For writing or using \(B(8, 0.05)\) |
| \(P(D \geq 2) = 1 - P(D \leq 1)\) | M1 | For writing or using \(1 - P(D \leq 1)\) |
| \(= 0.0572\) (calc 0.057244...) awrt 0.0572 | A1 | awrt 0.0572 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(E \sim Po(50)\) | M1 | For writing or using \(Po(50)\) |
| \(P(E=45) = \frac{e^{-50} \times 50^{45}}{45!}\) | M1 | For \(\frac{e^{-\lambda} \times \lambda^{45}}{45!}\) with any value of \(\lambda\) (may be implied by awrt 0.046) |
| \(= 0.0458262...\) awrt 0.0458 | A1 | awrt 0.0458 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(P(T > 16) = \frac{50-16}{50-10}\) or \(1 - \frac{16-10}{50-10}\) | M1 | For a correct method to find \(P(T > 16)\) |
| \(= 0.85\) | A1 | For 0.85 oe; correct answer scores 2 out of 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(P(T < 40) = 0.75\) | M1 | For 0.75 oe |
| \(F =\) number of customers ringing in next 40 seconds has \(F \sim Po(4)\) | ||
| \(P(F=0) [= e^{-4} =\) awrt \(0.0183]\) | M1 | For attempting \(P(F=0)\) from \(Po(\lambda)\), allow any \(\lambda\) |
| \(P(\text{Jia reaches correct dept and } F=0) = 0.75 \times 0.95 \times e^{-4}\) | dM1 | Dep on previous M1: "0.75"\(\times 0.95 \times\)"\(e^{-4}\)" |
| \(= 0.013049...\) awrt 0.013 | A1 | awrt 0.013 |
# Question 5:
## Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $D \sim B(8, 0.05)$ | M1 | For writing or using $B(8, 0.05)$ |
| $P(D \geq 2) = 1 - P(D \leq 1)$ | M1 | For writing or using $1 - P(D \leq 1)$ |
| $= 0.0572$ (calc 0.057244...) awrt **0.0572** | A1 | awrt 0.0572 |
## Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $E \sim Po(50)$ | M1 | For writing or using $Po(50)$ |
| $P(E=45) = \frac{e^{-50} \times 50^{45}}{45!}$ | M1 | For $\frac{e^{-\lambda} \times \lambda^{45}}{45!}$ with any value of $\lambda$ (may be implied by awrt 0.046) |
| $= 0.0458262...$ awrt **0.0458** | A1 | awrt 0.0458 |
## Part (c):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $P(T > 16) = \frac{50-16}{50-10}$ or $1 - \frac{16-10}{50-10}$ | M1 | For a correct method to find $P(T > 16)$ |
| $= 0.85$ | A1 | For 0.85 oe; correct answer scores 2 out of 2 |
## Part (d):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $P(T < 40) = 0.75$ | M1 | For 0.75 oe |
| $F =$ number of customers ringing in next 40 seconds has $F \sim Po(4)$ | | |
| $P(F=0) [= e^{-4} =$ awrt $0.0183]$ | M1 | For attempting $P(F=0)$ from $Po(\lambda)$, allow any $\lambda$ |
| $P(\text{Jia reaches correct dept and } F=0) = 0.75 \times 0.95 \times e^{-4}$ | dM1 | Dep on previous M1: "0.75"$\times 0.95 \times$"$e^{-4}$" |
| $= 0.013049...$ awrt **0.013** | A1 | awrt 0.013 |
---5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05
A random sample of 8 calls is taken.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt.
The receptionist receives 1000 calls each day.
\item Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day.
The total time, $T$ seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
\item Find $\mathrm { P } ( T > 16 )$
The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
\item Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2024 Q5 [12]}}