| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Conditional probability with geometric |
| Difficulty | Standard +0.8 This question requires understanding of geometric distributions and independence in part (i), verification in part (ii), then a sophisticated infinite series summation in part (iii). The key challenge is recognizing that P(R=S) = Σp²q^(2k) for k=0 to ∞, identifying this as a geometric series with ratio q², and simplifying the resulting expression. While the individual concepts are S1 standard, the multi-step algebraic manipulation and series summation for a probability result elevates this above typical routine exercises. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks |
|---|---|
| \(E^2\) | B1 1 |
| Answer | Marks |
|---|---|
| \((q-p)^2\) oe =AG | B1 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r=q^2\) | B1 | May be implied |
| \(\frac{a/(1-r)}{(S_\infty) = \frac{p^2}{1-q^2}}\) | M1 A1 | With \(a=p^2\)and \(r=q^2\)or \(q\) or \(q^4\) |
| \(= \frac{p^2}{1-(1-p)^2}\) \(p/(2-p)\) AG | M1 A1 5 | Attempt to simplify using \(p+q=1\) correctly. Dep on \(r = q^2\) or \(q^4\) \((1-q)^2 / (1-q)(1+q)\) or \(p^2/p(1+q)\) Correctly obtain given answer showing at least one intermediate step. |
### (i)
$E^2$ | B1 1 |
### (ii)
$(q-p)^2$ oe =AG | B1 1 |
### (iii)
$r=q^2$ | B1 | May be implied
$\frac{a/(1-r)}{(S_\infty) = \frac{p^2}{1-q^2}}$ | M1 A1 | With $a=p^2$and $r=q^2$or $q$ or $q^4$
$= \frac{p^2}{1-(1-p)^2}$ $p/(2-p)$ AG | M1 A1 5 | Attempt to simplify using $p+q=1$ correctly. Dep on $r = q^2$ or $q^4$ $(1-q)^2 / (1-q)(1+q)$ or $p^2/p(1+q)$ Correctly obtain given answer showing at least one intermediate step.
**P2Total [7]**
**Total 72 marks**$R$ and $S$ are independent random variables each having the distribution Geo$(p)$.
\begin{enumerate}[label=(\roman*)]
\item Find P$(R = 1$ and $S = 1)$ in terms of $p$. [1]
\item Show that P$(R = 3$ and $S = 3) = p^2q^4$, where $q = 1 - p$. [1]
\item Use the formula for the sum to infinity of a geometric series to show that
$$\text{P}(R = S) = \frac{p}{2-p}.$$ [5]
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2010 Q9 [7]}}