| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Poisson approximation justification or comparison |
| Difficulty | Standard +0.3 This is a straightforward application of Poisson distribution with clear parameters (n=1000000, p=1/200000 gives λ=5). Part (a) requires standard justification (large n, small p), parts (b)-(c) involve routine probability calculations using tables or calculators. The 10-second extension in (c) simply scales λ to 50. Slightly easier than average due to minimal conceptual challenge beyond recognizing the setup. |
| Spec | 5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02l Poisson conditions: for modelling5.02m Poisson: mean = variance = lambda |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | Nuclei decay randomly and decays are independent |
| Answer | Marks |
|---|---|
| a Poisson distribution is also appropriate | E1 |
| Answer | Marks |
|---|---|
| [3] | 2.4 |
| Answer | Marks |
|---|---|
| 2.4 | For partial explanation of binomial |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (b) | Po(5) |
| Answer | Marks |
|---|---|
| P(X >6)=1−0.762=0.238 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.3 |
| Answer | Marks |
|---|---|
| 1.1 | BC |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (c) | Mean = 10 × 5 = 50 |
| P(at least 60 decays)=1−0.9077=0.0923 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 3.3 | |
| 1.1 | BC | Allow 0.092 |
Question 4:
4 | (a) | Nuclei decay randomly and decays are independent
with constant probability 1
200000
The number of decays out of 1000000 is being
counted, so a binomial distribution is appropriate
Because n = 1000000 is large and p= 1 is small
200000
a Poisson distribution is also appropriate | E1
E1
E1
[3] | 2.4
2.4
2.4 | For partial explanation of binomial
For full explanation
For explanation of Poisson
4 | (b) | Po(5)
P(X =6)=0.146
P(X >6)=1−0.762=0.238 | M1
A1
A1
[3] | 3.3
1.1
1.1 | BC
BC
4 | (c) | Mean = 10 × 5 = 50
P(at least 60 decays)=1−0.9077=0.0923 | B1
B1
[2] | 3.3
1.1 | BC | Allow 0.0924 A radioactive source contains 1000000 nuclei of a particular radioisotope. On average 1 in 200000 of these nuclei will decay in a period of 1 second. The random variable $X$ represents the number of nuclei which decay in a period of 1 second. You should assume that nuclei decay randomly and independently of each other.
\begin{enumerate}[label=(\alph*)]
\item Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of $X$.
Use a Poisson distribution to answer parts (b) and (c).
\item Calculate each of the following probabilities.
\begin{itemize}
\item $\mathrm { P } ( X = 6 )$
\item $\mathrm { P } ( X > 6 )$
\item Determine an estimate of the probability that at least 60 nuclei decay in a period of 10 seconds.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q4 [8]}}