| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find minimum n for P(X ≤ n) > threshold |
| Difficulty | Moderate -0.3 This is a straightforward application of the geometric distribution formula with minimal problem-solving required. Part (i) involves a simple calculation of P(X ≤ 4) = 1 - (5/6)^4, while part (ii) requires solving 1 - (5/6)^N > 0.95 using logarithms—both are standard textbook exercises testing basic understanding rather than insight or extended reasoning. |
| Spec | 5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| 6(i) | P(X ≤ 4) = 1 – q4 | M1 |
| = 671/1296 or 0⋅518 | A1 | Set q = 5/6 and evaluate |
| Total: | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(ii) | 1 – qN–1 > 0⋅95 | M1 |
| (5/6)N –1 < 0⋅05, N – 1 > log 0⋅05 / log 5/6 | M1 | Set q = 5/6, rearrange and take logs (any base) to give |
| Answer | Marks | Guidance |
|---|---|---|
| min | A1 | Find N |
| Answer | Marks |
|---|---|
| Total: | 3 |
Question 6:
--- 6(i) ---
6(i) | P(X ≤ 4) = 1 – q4 | M1 | Find prob. of score of 6 on no more than 4 throws
= 671/1296 or 0⋅518 | A1 | Set q = 5/6 and evaluate
Total: | 2
--- 6(ii) ---
6(ii) | 1 – qN–1 > 0⋅95 | M1 | Formulate condition for N (1 – qN is M0)
(5/6)N –1 < 0⋅05, N – 1 > log 0⋅05 / log 5/6 | M1 | Set q = 5/6, rearrange and take logs (any base) to give
bound
N – 1 > 16⋅4[3], N = 18
min | A1 | Find N
min
(N – 1 < 16⋅4 or N – 1 = 16⋅4 earns M1 M1 A0)
Total: | 3A fair die is thrown repeatedly until a 6 is obtained.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that obtaining a 6 takes no more than four throws. [2]
\item Find the least integer $N$ such that the probability of obtaining a 6 before the $N$th throw is more than 0.95. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2017 Q6 [5]}}