| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Sampling without replacement until success |
| Difficulty | Standard +0.3 This is a straightforward S1 question on sampling without replacement with clear structure and standard calculations. Part (a) guides students through the method, parts (b-c) are routine probability distribution work, and parts (d-e) apply expected value in a simple context. The calculations are manageable and the question type is common in textbooks, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{4}{6} \times \frac{2}{5} = \frac{4}{15}\) | M1 A1 | |
| (b) Same method, giving | M2 A2 | |
| \(b\) | 1 | 2 |
| \(P(B = b)\) | \(\frac{1}{3}\) | \(\frac{4}{15}\) |
| (c) \(\sum bP(b) = \frac{1}{15}(5 + 8 + 9 + 8 + 5) = \frac{35}{15} = \frac{7}{3}\) | M2 A1 | |
| (d) \(P(\text{winning}) = \frac{1}{3} + \frac{4}{15} = \frac{3}{5}\) | M1 A1 | |
| expected winnings \(= \frac{3}{5} \times 50 = 30\) pence | M1 A1 | |
| (e) \((3 \times 30) - 100 = -10 \therefore\) 10 pence loss | M2 A1 | (16 marks total) |
(a) $\frac{4}{6} \times \frac{2}{5} = \frac{4}{15}$ | M1 A1 |
(b) Same method, giving | M2 A2 |
| $b$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $P(B = b)$ | $\frac{1}{3}$ | $\frac{4}{15}$ | $\frac{1}{5}$ | $\frac{2}{15}$ | $\frac{1}{15}$ |
(c) $\sum bP(b) = \frac{1}{15}(5 + 8 + 9 + 8 + 5) = \frac{35}{15} = \frac{7}{3}$ | M2 A1 |
(d) $P(\text{winning}) = \frac{1}{3} + \frac{4}{15} = \frac{3}{5}$ | M1 A1 |
expected winnings $= \frac{3}{5} \times 50 = 30$ pence | M1 A1 |
(e) $(3 \times 30) - 100 = -10 \therefore$ 10 pence loss | M2 A1 | (16 marks total)
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**Total: 75 marks**7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag.
The random variable $B$ is the number of balls that have been removed from the bag.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }$.
\item Find the probability distribution of $B$.
\item Find $\mathrm { E } ( B )$.
The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
\item What are the expected winnings from playing this game once?
For $\pounds 1$, a contestant gets to play the game three times.
\item What is the expected profit or loss from the three games?
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q7 [16]}}