| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Geometric with multiple success milestones |
| Difficulty | Standard +0.3 This is a straightforward S1 binomial/geometric probability question. Part (i)(a) requires recognizing that X=10 means 10 steps right then 1 up, giving (0.8)^10(0.2). Part (i)(b) sums probabilities for X<10. Parts (ii) and (iii) use standard expectation formulas for geometric distributions. All techniques are routine for S1 with no novel insight required, making it slightly easier than average. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
### (ii)
**Answer:** $\frac{1}{0.2}$ alone = 5
**Marks:** M1, A1[2]
**Guidance:**
- Allow 1 ÷ their incorrect $p$ used in (i)(a)
- Ignore eg "E(X)="
### (iii)
**Answer:** 4 Allow (4, 1)
**Marks:** B1ft[1]
**Guidance:**
- or (ii) − 1 or (ii) × 0.8
- If 1 + 0.8 = 1.25, see above: If 1 + 0.8 = 1.25, see aboveA game is played with a token on a board with a grid printed on it. The token starts at the point $(0, 0)$ and moves in steps. Each step is either 1 unit in the positive $x$-direction with probability 0.8, or 1 unit in the positive $y$-direction with probability 0.2. The token stops when it reaches a point with a $y$-coordinate of 1. It is given that the token stops at $(X, 1)$.
\begin{enumerate}[label=(\roman*)]
\item \begin{enumerate}[label=(\alph*)]
\item Find the probability that $X = 10$. [2]
\item Find the probability that $X < 10$. [3]
\end{enumerate}
\item Find the expected number of steps taken by the token. [2]
\item Hence, write down the value of E$(X)$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2013 Q9 [8]}}