| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Compound event with two dice/coins |
| Difficulty | Moderate -0.8 This is a straightforward application of geometric distribution with clearly defined success probability (1/4 for pair of heads). Part (i) requires direct substitution into the geometric probability formula, part (ii) requires summing a short geometric series or using the CDF. The conceptual setup is simple and the calculations are routine for Further Maths students. |
| Spec | 5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(E(X) = 1/p = 1/\frac{1}{4} = 4\) | B1 | State or find the expected value of \(X\) using \(p = \frac{1}{4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X=4) = (\frac{3}{4})^3\frac{1}{4} = 27/256\) or \(0.105\) | M1 A1 | Find \(P(X=4)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(X<6) = 1 - (\frac{3}{4})^5\) | ||
| or \(\{1 + \frac{3}{4} + (\frac{3}{4})^2 + (\frac{3}{4})^3 + (\frac{3}{4})^4\}\frac{1}{4}\) | ||
| \(= 781/1024\) or \(0.763\) | M1 A1 | S.R. Using \(p = \frac{1}{2}\) can earn B0 M1 A0 M0 A0 |
# Question 6:
| $E(X) = 1/p = 1/\frac{1}{4} = 4$ | B1 | State or find the expected value of $X$ using $p = \frac{1}{4}$ |
**Part (i):**
| $P(X=4) = (\frac{3}{4})^3\frac{1}{4} = 27/256$ or $0.105$ | M1 A1 | Find $P(X=4)$ |
**Part (ii):**
| $P(X<6) = 1 - (\frac{3}{4})^5$ | | |
| or $\{1 + \frac{3}{4} + (\frac{3}{4})^2 + (\frac{3}{4})^3 + (\frac{3}{4})^4\}\frac{1}{4}$ | | |
| $= 781/1024$ or $0.763$ | M1 A1 | S.R. Using $p = \frac{1}{2}$ can earn B0 M1 A0 M0 A0 |
---6 A pair of coins is thrown repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable $X$. State the expected value of $X$.
Find the probability that\\
(i) exactly 4 throws are required to obtain a pair of heads,\\
(ii) fewer than 6 throws are required to obtain a pair of heads.
\hfill \mbox{\textit{CAIE FP2 2014 Q6 [5]}}