| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.3 This is a straightforward S1 probability question requiring enumeration of outcomes for independent events, constructing a probability distribution, and calculating expectation/variance using standard formulas. Part (a) is scaffolded (showing a given answer), and while there are multiple parts totaling 16 marks, each component uses routine techniques without requiring problem-solving insight or novel approaches. Slightly easier than average due to the structured nature and standard methods. |
| Spec | 2.03b Probability diagrams: tree, Venn, sample space5.02b Expectation and variance: discrete random variables5.02c Linear coding: effects on mean and variance |
| Answer | Marks |
|---|---|
| (a) \((0.6 \times 0.5 \times 0.7) + (0.6 \times 0.5 \times 0.3) + (0.4 \times 0.5 \times 0.3) = 0.36\) | M3 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(w\) | 0 | 1 |
| \(P(W=w)\) | 0.14 | 0.41 |
| (c) \(E(W) = \sum w P(w) = 0 + 0.41 + 0.72 + 0.27 = 1.4\) | M1 A1 | |
| \(E(W^2) = \sum w^2 P(w) = 0 + 0.41 + 1.44 + 0.81 = 2.66\) | M1 A1 | |
| \(\text{Var}(W) = 2.66 - 1.4^2 = 0.7\) | M1 A1 | |
| (d) e.g. unlikely to be valid as result of each match will probably raise or lower confidence changing probability of success in the next match | B2 | (16 marks total) |
**(a)** $(0.6 \times 0.5 \times 0.7) + (0.6 \times 0.5 \times 0.3) + (0.4 \times 0.5 \times 0.3) = 0.36$ | M3 A1 |
**(b)** $P(W=0) = 0.4 \times 0.5 \times 0.7 = 0.14$
$P(W=3) = 0.6 \times 0.5 \times 0.3 = 0.09$
$P(W=1) = 1 - (0.14 + 0.36 + 0.09) = 0.41$
| $w$ | 0 | 1 | 2 | 3 |
| $P(W=w)$ | 0.14 | 0.41 | 0.36 | 0.09 | | M2 A2 |
**(c)** $E(W) = \sum w P(w) = 0 + 0.41 + 0.72 + 0.27 = 1.4$ | M1 A1 |
$E(W^2) = \sum w^2 P(w) = 0 + 0.41 + 1.44 + 0.81 = 2.66$ | M1 A1 |
$\text{Var}(W) = 2.66 - 1.4^2 = 0.7$ | M1 A1 |
**(d)** e.g. unlikely to be valid as result of each match will probably raise or lower confidence changing probability of success in the next match | B2 | **(16 marks total)**
---A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.
\begin{enumerate}[label=(\alph*)]
\item Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]
\end{enumerate}
Let the random variable $W$ be the number of matches that the team win in the league.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the probability distribution of $W$. [4 marks]
\item Find E$(W)$ and Var$(W)$. [6 marks]
\item Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q5 [16]}}