| Exam Board | Edexcel |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson with binomial combination |
| Difficulty | Standard +0.3 This is a straightforward multi-part Poisson distribution question requiring standard techniques: scaling the rate parameter, using tables/calculator for probabilities, and applying binomial-to-Poisson approximation. Part (b) is verification only. The most challenging aspect is recognizing the binomial-to-Poisson setup in part (c), but this is a standard Further Stats 1 technique with clear signposting. Slightly above average difficulty due to the multi-stage structure and approximation requirement, but all steps are routine for FS1 students. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(X \sim \text{Po}(7.2)\) | M1 | Writing or using Po(7.2) |
| \(P(X=7) = 0.14858\ldots\) awrt 0.149 | A1 | awrt 0.149 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(Y \sim \text{Po}(19.2)\) | M1 | Writing or using Po(19.2) |
| \(P(Y>28) = 1 - P(Y \leq 28) = 1 - 0.9780\ldots = 0.02199\ldots\) 0.022 | A1* | cso given answer with correct probability statement e.g. \(1-P(Y \leq 28)\) and no incorrect working seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(100 \times 0.022\) awrt 2.2 | B1 | awrt 2.2 (isw once awrt 2.2 is seen) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{100(0.022)(1-0.022)}\) | M1 | Correct expression including square root |
| \(= 1.466\ldots\) awrt 1.47 | A1 | awrt 1.47. Note: \(\sqrt{2.2} = 1.483\ldots\) is M0A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(B(100, 0.022) \rightarrow \text{Po}(2.2)\) | M1 | Approximating binomial (100, 0.022) with Po(2.2) [may be seen in (i) or (ii)] |
| \(P(W \geq 6) = 1 - P(W \leq 5) = 1 - 0.9750\ldots\) | M1 | Using \(1 - P(W \leq 5)\) from Poisson distribution |
| \(= 0.02490\ldots\) awrt 0.0249 | A1 | awrt 0.0249. Note: Using Binomial \(1 - 0.9765588\ldots = 0.02344\ldots\) scores M0M0A0 |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $X \sim \text{Po}(7.2)$ | M1 | Writing or using Po(7.2) |
| $P(X=7) = 0.14858\ldots$ awrt **0.149** | A1 | awrt 0.149 |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $Y \sim \text{Po}(19.2)$ | M1 | Writing or using Po(19.2) |
| $P(Y>28) = 1 - P(Y \leq 28) = 1 - 0.9780\ldots = 0.02199\ldots$ **0.022** | A1* | cso given answer with correct probability statement e.g. $1-P(Y \leq 28)$ and no incorrect working seen |
## Part (c)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $100 \times 0.022$ awrt **2.2** | B1 | awrt 2.2 (isw once awrt 2.2 is seen) |
## Part (c)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{100(0.022)(1-0.022)}$ | M1 | Correct expression including square root |
| $= 1.466\ldots$ awrt **1.47** | A1 | awrt 1.47. Note: $\sqrt{2.2} = 1.483\ldots$ is M0A0 |
## Part (c)(iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $B(100, 0.022) \rightarrow \text{Po}(2.2)$ | M1 | Approximating binomial (100, 0.022) with Po(2.2) [may be seen in (i) or (ii)] |
| $P(W \geq 6) = 1 - P(W \leq 5) = 1 - 0.9750\ldots$ | M1 | Using $1 - P(W \leq 5)$ from Poisson distribution |
| $= 0.02490\ldots$ awrt **0.0249** | A1 | awrt 0.0249. Note: Using Binomial $1 - 0.9765588\ldots = 0.02344\ldots$ scores M0M0A0 |
---\begin{enumerate}
\item A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.\\
(a) Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day.
\end{enumerate}
The company has enough staff to respond to 28 call-outs in an 8 -hour working day.\\
(b) Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places.
In a random sample of 100 working days each of 8 hours,\\
(c) (i) find the expected number of days that the company receives more than 28 call-outs,\\
(ii) find the standard deviation of the number of days that the company receives more than 28 call-outs,\\
(iii) use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.
\hfill \mbox{\textit{Edexcel FS1 AS 2020 Q1 [10]}}