| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Compound event with two dice/coins |
| Difficulty | Moderate -0.3 This is a straightforward application of the geometric distribution with clearly defined success probability p=1/36. Parts (i)-(iii) require only direct formula substitution: E(X)=1/p for the mean, P(X=12)=(35/36)^11(1/36) for exactly 12 throws, and P(X>12)=(35/36)^12 for more than 12 throws. No problem-solving insight or multi-step reasoning is needed—just recognition of the geometric distribution and recall of standard formulas, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(p = (1/6)^2\) or \(1/36\) | B1 | Find (or imply) probability \(p\) of pair of sixes in one throw |
| \(1/p = 36\) | B1 | Find mean value of \(X\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=12) = p(1-p)^{11} = 0.0204\) | M1 A1 | Find prob. of needing exactly 12 throws |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X > 12) = (1-p)^{12} = 0.713\) | M1 A1 | Find prob. of needing more than 12 throws |
## Question 6(i):
$p = (1/6)^2$ or $1/36$ | B1 | Find (or imply) probability $p$ of pair of sixes in one throw
$1/p = 36$ | B1 | Find mean value of $X$
## Question 6(ii):
$P(X=12) = p(1-p)^{11} = 0.0204$ | M1 A1 | Find prob. of needing exactly 12 throws
## Question 6(iii):
$P(X > 12) = (1-p)^{12} = 0.713$ | M1 A1 | Find prob. of needing more than 12 throws
**Total: 2**
---6 A pair of fair dice is thrown repeatedly until a pair of sixes is obtained. The number of throws taken is denoted by the random variable $X$.\\
(i) Find the mean value of $X$.\\
(ii) Find the probability that exactly 12 throws are required to obtain a pair of sixes.\\
(iii) Find the probability that more than 12 throws are required to obtain a pair of sixes.\\
\hfill \mbox{\textit{CAIE FP2 2017 Q6 [6]}}