CAIE FP2 2013 November — Question 6 6 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2013
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeometric Distribution
TypeFind minimum n for P(X ≤ n) > threshold
DifficultyStandard +0.3 This is a straightforward application of geometric distribution with clearly defined success probability (1/3). Parts involve standard formulas: mean = 1/p, P(X > n) = (1-p)^n, and solving an inequality with logarithms. While it requires multiple steps and understanding of the geometric distribution, all techniques are routine for Further Maths students with no novel problem-solving required.
Spec5.02f Geometric distribution: conditions5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)
6 A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by the random variable \(X\). State the mean value of \(X\). Find the probability that obtaining a 5 or a 6 takes more than 8 throws. Find the least integer \(n\) such that the probability of obtaining a 5 or a 6 in fewer than \(n\) throws is more than 0.99.
Question 6:
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(E(X) = 1/\frac{1}{3} = 3\)B1 State or find the mean of \(X\)
\(P(X > 8) = (\frac{2}{3})^8 = 256/6561\) or \(0.0390\)M1 A1 Find \(P(X > 8)\)
\(1 - (\frac{2}{3})^{n-1} > 0.99\), \((\frac{2}{3})^{n-1} < 0.01\)M1 Formulate condition for \(n\) (M0 if equality used)
\(n - 1 > \log 0.01 / \log \frac{2}{3}\)M1 Take logs (any base) to give inequality for \(n\)
\(n - 1 > 11.4\), \(n_{min} = 13\)A1 Find \(n_{min}\)
Total: 6 marks 
## Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $E(X) = 1/\frac{1}{3} = 3$ | B1 | State or find the mean of $X$ |
| $P(X > 8) = (\frac{2}{3})^8 = 256/6561$ or $0.0390$ | M1 A1 | Find $P(X > 8)$ |
| $1 - (\frac{2}{3})^{n-1} > 0.99$, $(\frac{2}{3})^{n-1} < 0.01$ | M1 | Formulate condition for $n$ (M0 if equality used) |
| $n - 1 > \log 0.01 / \log \frac{2}{3}$ | M1 | Take logs (any base) to give inequality for $n$ |
| $n - 1 > 11.4$, $n_{min} = 13$ | A1 | Find $n_{min}$ |
| **Total: 6 marks** | | |

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6 A fair die is thrown until a 5 or a 6 is obtained. The number of throws taken is denoted by the random variable $X$. State the mean value of $X$.

Find the probability that obtaining a 5 or a 6 takes more than 8 throws.

Find the least integer $n$ such that the probability of obtaining a 5 or a 6 in fewer than $n$ throws is more than 0.99.

\hfill \mbox{\textit{CAIE FP2 2013 Q6 [6]}}
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