| Exam Board | Edexcel |
|---|---|
| Module | FS1 (Further Statistics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Distribution |
| Type | Game theory with alternating players |
| Difficulty | Standard +0.8 This is a multi-part Further Statistics question requiring understanding of geometric distribution in an alternating-player context. Parts (a)-(c) are standard geometric distribution applications, but part (d) requires recognizing an infinite geometric series for alternating probabilities. The conceptual leap to model alternating turns and sum the infinite series elevates this above routine exercises, though the individual techniques are well-practiced at this level. |
| Spec | 5.02b Expectation and variance: discrete random variables5.02g Geometric probabilities: P(X=r) = p(1-p)^(r-1)5.02h Geometric: mean 1/p and variance (1-p)/p^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([X \sim \text{Geo}(0.2)]\) Suzanne's 4th selection is the 7th overall | M1 | Selecting geometric distribution with \(p = 0.2\); allow \((0.8)^n(0.2)\) with \(n=6\) or \(n=3\) |
| \(P(X=7) = (0.8)^6(0.2)\) or \((0.64)^3(0.2)\) | ||
| \(= 0.05242...\) awrt 0.0524 | A1 | Allow exact fraction \(\frac{4096}{78125}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(X \geq 6) [= (1-0.2)^5]\) | M1 | May be implied by \((1-p)^5\) or \(1-(p+pq+pq^2+pq^3+pq^4)\) |
| \(= 0.32768\) awrt 0.328 | A1 | Allow exact fraction \(\frac{1024}{3125}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Mean \(= 5\) | B1 | |
| Standard deviation \(= \sqrt{\frac{1-0.2}{0.2^2}} = \sqrt{20}\) awrt 4.47 | B1 | \(\sqrt{20}\) o.e. or awrt 4.47 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(P(\text{Suzanne wins}) = 0.2 + (0.8)^2(0.2) + (0.8)^4(0.2) + \ldots\) | M1 | Determining probability Suzanne wins with at least three terms seen |
| Infinite geometric series \(= \frac{0.2}{1-0.8^2}\) (oe) | M1 | Recognising need to sum infinite geometric series with correct \(r = 0.8^2\) |
| \(= \frac{5}{9}\) | A1 | Allow awrt 0.556 |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[X \sim \text{Geo}(0.2)]$ Suzanne's 4th selection is the 7th overall | M1 | Selecting geometric distribution with $p = 0.2$; allow $(0.8)^n(0.2)$ with $n=6$ or $n=3$ |
| $P(X=7) = (0.8)^6(0.2)$ or $(0.64)^3(0.2)$ | | |
| $= 0.05242...$ awrt **0.0524** | A1 | Allow exact fraction $\frac{4096}{78125}$ |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(X \geq 6) [= (1-0.2)^5]$ | M1 | May be implied by $(1-p)^5$ or $1-(p+pq+pq^2+pq^3+pq^4)$ |
| $= 0.32768$ awrt **0.328** | A1 | Allow exact fraction $\frac{1024}{3125}$ |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Mean $= 5$ | B1 | |
| Standard deviation $= \sqrt{\frac{1-0.2}{0.2^2}} = \sqrt{20}$ awrt **4.47** | B1 | $\sqrt{20}$ o.e. or awrt 4.47 |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $P(\text{Suzanne wins}) = 0.2 + (0.8)^2(0.2) + (0.8)^4(0.2) + \ldots$ | M1 | Determining probability Suzanne wins with at least three terms seen |
| Infinite geometric series $= \frac{0.2}{1-0.8^2}$ (oe) | M1 | Recognising need to sum infinite geometric series with correct $r = 0.8^2$ |
| $= \frac{5}{9}$ | A1 | Allow awrt 0.556 |
---\begin{enumerate}
\item Suzanne and Jon are playing a game.
\end{enumerate}
They put 4 red counters and 1 blue counter in a bag.\\
Suzanne reaches into the bag and selects one of the counters at random. If the counter she selects is blue, she wins the game. Otherwise she puts it back in the bag and Jon selects one at random. If the counter he selects is blue, he wins the game. Otherwise he puts it back in the bag and they repeat this process until one of them selects the blue counter.\\
(a) Find the probability that Suzanne selects the blue counter on her 4th selection.\\
(b) Find the probability that the blue counter is first selected on or after Jon's third selection.\\
(c) Find the mean and standard deviation of the number of selections made until the blue counter is selected.\\
(d) Find the probability that Suzanne wins the game.
\hfill \mbox{\textit{Edexcel FS1 2020 Q3 [9]}}