CAIE FP2 2016 June — Question 6 5 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2016
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicGeometric Distribution
TypeFind minimum n for P(X ≤ n) > threshold
DifficultyStandard +0.3 This question requires identifying the probability of success (p=5/36 for getting sum of 6 with two dice), then applying standard geometric distribution formulas for mean (1/p) and cumulative probability. The second part involves solving an inequality with logarithms, which is routine for Further Maths students. All steps are straightforward applications of known results.
Spec5.02f 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
6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable \(X\). Find the mean of \(X\). Find the least integer \(N\) such that the probability of obtaining a score of 6 in fewer than \(N\) throws is more than 0.95 .
Question 6:
Part 1 (mean of \(X\)):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(p = 5/36\) or \(0.139\)B1 Find probability \(p\) of score of 6 on one throw
\(1/p = 36/5\) or \(7.2\)B1 Find mean of \(X\); Total: [2]
Part 2 (finding \(N_{\min}\)):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(1 - q^{N-1} > 0.95\)M1 Formulate condition for \(N(1 - q^N\) is M0)
\((31/36)^{N-1} < 0.05\)M1 Rearrange and take logs (any base) to give bound
\(N - 1 > \log 0.05 / \log 31/36\)
\(N - 1 > 20.03\), \(N_{\min} = 22\)A1 Find \(N_{\min}\); (\(N-1 < 20.03\) or \(N-1 = 20.03\) earns M1 M1 A0); Total: [3] 
## Question 6:

### Part 1 (mean of $X$):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $p = 5/36$ or $0.139$ | B1 | Find probability $p$ of score of 6 on one throw |
| $1/p = 36/5$ or $7.2$ | B1 | Find mean of $X$; Total: [2] |

### Part 2 (finding $N_{\min}$):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1 - q^{N-1} > 0.95$ | M1 | Formulate condition for $N(1 - q^N$ is M0) |
| $(31/36)^{N-1} < 0.05$ | M1 | Rearrange and take logs (any base) to give bound |
| $N - 1 > \log 0.05 / \log 31/36$ | | |
| $N - 1 > 20.03$, $N_{\min} = 22$ | A1 | Find $N_{\min}$; ($N-1 < 20.03$ or $N-1 = 20.03$ earns M1 M1 A0); Total: [3] |

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6 The score when two fair dice are thrown is the sum of the two numbers on the upper faces. Two fair dice are thrown repeatedly until a score of 6 is obtained. The number of throws taken is denoted by the random variable $X$. Find the mean of $X$.

Find the least integer $N$ such that the probability of obtaining a score of 6 in fewer than $N$ throws is more than 0.95 .

\hfill \mbox{\textit{CAIE FP2 2016 Q6 [5]}}
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