| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Joint probability of separate processes |
| Difficulty | Moderate -0.8 Part (a) requires simple recall of standard Poisson conditions (independence and constant rate). Parts (b) and (c) are routine calculations using Poisson distribution properties and scaling, with (c) requiring recognition that two stretches follow Po(28.5) and basic probability manipulation. No novel insight or complex problem-solving needed. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Flaws occur independently of one another | B1 | At least one contextualised reason, ignore "singly" |
| and at constant average rate (or "uniform rate" which is equivalent) | B1[2] | A second assumption, not "singly". Allow "fixed average rate". *Not*: "constant rate" or "average constant rate" or "constant average number" or "constant probability" or "probability same for each 1 km". More than 2 different assumptions (ignore "singly"): max B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(\leq 11) - P(\leq 7)\) | M1 | Allow M1 for 0.109(24)... *Not*, e.g. \(P(\leq 11) - (1 - P(\leq 7)) = 0.77\) |
| \(= 0.202\ (0.20171)\) | A1[2] | Awrt 0.202 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(Po(5.7 \times 5)\ (= Po(28.5))\) | M1 | Stated or implied |
| *their* \(P(\leq 29) \times (1 - \text{their}\ P(\leq 29)) \times 2\ (= 0.586 \times 0.414 \times 2)\) | M1 | Or \(1 - [\text{their}\ P(\leq 29)]^2 - [1 - \text{their}\ P(\leq 29)]^2\ (= 1 - 0.586^2 - 0.414^2)\). Allow M1 for 2 omitted or e.g. \(P(\leq 29) \times (1 - P(\leq 30)) \times 2\) (both: M0) |
| \(= 0.485\ (19)\) | A1[3] | Awrt 0.485 |
## Question 1:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Flaws occur independently of one another | B1 | At least one contextualised reason, ignore "singly" |
| and at constant average rate (or "uniform rate" which is equivalent) | B1[2] | A second assumption, not "singly". Allow "fixed average rate". *Not*: "constant rate" or "average constant rate" or "constant average number" or "constant probability" or "probability same for each 1 km". More than 2 different assumptions (ignore "singly"): max B1 |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(\leq 11) - P(\leq 7)$ | M1 | Allow M1 for 0.109(24)... *Not*, e.g. $P(\leq 11) - (1 - P(\leq 7)) = 0.77$ |
| $= 0.202\ (0.20171)$ | A1[2] | Awrt 0.202 |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $Po(5.7 \times 5)\ (= Po(28.5))$ | M1 | Stated or implied |
| *their* $P(\leq 29) \times (1 - \text{their}\ P(\leq 29)) \times 2\ (= 0.586 \times 0.414 \times 2)$ | M1 | Or $1 - [\text{their}\ P(\leq 29)]^2 - [1 - \text{their}\ P(\leq 29)]^2\ (= 1 - 0.586^2 - 0.414^2)$. Allow M1 for 2 omitted or e.g. $P(\leq 29) \times (1 - P(\leq 30)) \times 2$ (both: M0) |
| $= 0.485\ (19)$ | A1[3] | Awrt 0.485 |
---1 A radar device is used to detect flaws in motorway roads before they become dangerous. The number of flaws in a 1 km stretch of motorway is denoted by $X$. It may be assumed that flaws occur randomly.
\begin{enumerate}[label=(\alph*)]
\item State two further assumptions that are necessary for $X$ to be well modelled by a Poisson distribution.
Assume now that $X$ can be modelled by distribution $\operatorname { Po } ( 5.7 )$.
\item Determine the probability that in a randomly chosen stretch of motorway, of length 1 km , there are between 8 and 11 flaws, inclusive.
\item Determine the probability that in two randomly chosen, non-overlapping, stretches of motorway, each of length 5 km , there are at least 30 flaws in one stretch and fewer than 30 flaws in the other stretch.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2023 Q1 [7]}}