| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Compound event with two dice/coins |
| Difficulty | Standard +0.3 This is a straightforward application of geometric distribution formulas. Part (a) uses P(X > n) = (1-p)^n with p=1/6. Part (b) requires recognizing the new success probability is 1/36 and applying E(X)=1/p. Part (c) uses the geometric PMF twice. All parts are direct formula application with no conceptual challenges beyond identifying the correct probability for paired dice. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(\frac{5}{6}\right)^8\) | M1 | \(p^8\), \(0 < p < 1\), no \(x\), no + or − |
| 0.233 | A1 |
| Answer | Marks |
|---|---|
| 36 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=10) + P(X=11) = \left(\frac{35}{36}\right)^9 \frac{1}{36} + \left(\frac{35}{36}\right)^{10} \frac{1}{36}\) | M1 | OE, unsimplified expression in form \(p^9q + p^{10}q\), \(p + q = 1\), no \(\times\) |
| 0.0425 | A1 |
## Question 2:
### Part 2(a):
$\left(\frac{5}{6}\right)^8$ | M1 | $p^8$, $0 < p < 1$, no $x$, no + or −
0.233 | A1 |
### Part 2(b):
36 | B1 |
### Part 2(c):
$P(X=10) + P(X=11) = \left(\frac{35}{36}\right)^9 \frac{1}{36} + \left(\frac{35}{36}\right)^{10} \frac{1}{36}$ | M1 | OE, unsimplified expression in form $p^9q + p^{10}q$, $p + q = 1$, no $\times$
0.0425 | A1 |
---2 An ordinary fair die is thrown until a 6 is obtained.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that obtaining a 6 takes more than 8 throws.\\
Two ordinary fair dice are thrown together until a pair of 6s is obtained. The number of throws taken is denoted by the random variable $X$.
\item Find the expected value of $X$.
\item Find the probability that obtaining a pair of 6s takes either 10 or 11 throws.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q2 [5]}}