| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | 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 identification of p=1/4, then routine calculations of P(X=3) and P(X<4), followed by solving an inequality involving (3/4)^N < 0.05. All parts use standard formulas with no conceptual challenges beyond recognizing the setup, making it slightly easier than average for Further Maths. |
| 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 |
7 A pair of fair coins is thrown repeatedly until a pair of tails is obtained. The number of throws taken is denoted by the random variable $X$.\\
(i) State the expected value of $X$.\\
(ii) Find the probability that exactly 3 throws are required to obtain a pair of tails.\\
(iii) Find the probability that fewer than 4 throws are required to obtain a pair of tails.\\
(iv) Find the least integer $N$ such that the probability of obtaining a pair of tails in fewer than $N$ throws is more than 0.95 .\\
\hfill \mbox{\textit{CAIE FP2 2019 Q7 [8]}}