| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Find minimum n for P(X ≤ n) > threshold |
| Difficulty | Standard +0.3 This is a straightforward geometric distribution question requiring standard formula application. Part (i) establishes p=2/3 using mean=1/p and variance=(1-p)/p², then parts (ii-iv) involve direct substitution into geometric distribution formulas. The algebra is simple and all steps are routine for Further Maths students, 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 |
A biased coin is tossed repeatedly until a head is obtained. The random variable $X$ denotes the number of tosses required for a head to be obtained. The mean of $X$ is equal to twice the variance of $X$. Show that the probability that a head is obtained when the coin is tossed once is $\frac{2}{3}$. [2]
Find
\begin{enumerate}[label=(\roman*)]
\item P($X = 4$), [1]
\item P($X > 4$), [2]
\item the least integer $N$ such that P($X \leq N$) $> 0.999$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2015 Q6 [8]}}