| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 8 |
| 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 geometric distribution formulas with p=1/6. Parts (i) and (ii) require direct substitution into standard formulas, while part (iii) involves solving an inequality using logarithms—all routine techniques for Further Maths students with no novel problem-solving required. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Find probability for needing 5 throws: \(p(1-p)^4\) with \(p = 1/6;\ = 0{\cdot}0804\) | M1 A1; A1 | [Subtotal: 3] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Find probability for needing \(< 8\) throws: \(1 - (1-p)^7 = 0{\cdot}721\) | M1 A1 | [Subtotal: 2] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Relate prob. to 0.99 (allow \(>\) but not \(=\)): \(1 - (1-p)^{n-1} \geq 0{\cdot}99\) | B1 | |
| Find least integer \(n\): \((n-1)\log\frac{5}{6} \leq \log 0{\cdot}01\) | M1 | |
| (Allow M1 A1 even if equality used): \(n - 1 \geq 25{\cdot}3,\ n_{min} = 27\) | A1 | [Subtotal: 3] |
## Question 7:
### Part (i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Find probability for needing 5 throws: $p(1-p)^4$ with $p = 1/6;\ = 0{\cdot}0804$ | M1 A1; A1 | [Subtotal: 3] |
### Part (ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Find probability for needing $< 8$ throws: $1 - (1-p)^7 = 0{\cdot}721$ | M1 A1 | [Subtotal: 2] |
### Part (iii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Relate prob. to 0.99 (allow $>$ but not $=$): $1 - (1-p)^{n-1} \geq 0{\cdot}99$ | B1 | |
| Find least integer $n$: $(n-1)\log\frac{5}{6} \leq \log 0{\cdot}01$ | M1 | |
| (Allow M1 A1 even if equality used): $n - 1 \geq 25{\cdot}3,\ n_{min} = 27$ | A1 | [Subtotal: 3] |
**Total: 8**
---7 A fair die is thrown until a 6 appears for the first time. Assuming that the throws are independent, find\\
(i) the probability that exactly 5 throws are needed,\\
(ii) the probability that fewer than 8 throws are needed,\\
(iii) the least integer $n$ such that the probability of obtaining a 6 before the $n$th throw is at least 0.99 .
\hfill \mbox{\textit{CAIE FP2 2011 Q7 [8]}}