| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Sampling without replacement |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic probability distribution concepts. Part (a) is simple recall, part (b) requires routine hypergeometric probability calculations with small numbers, and part (c) applies standard formulas for expectation and variance. All techniques are textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 1 | 2 |
| P\((X = x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) A quantity which can take only certain distinct values, with fixed probabilities | B1 | |
| (b) | x | 0 |
| P(X = x) | \(\frac{1}{7}\) | \(\frac{4}{7}\) |
| (c) (i) \(E(X) = \frac{8}{7}\) | B1; M1 A1 | |
| (ii) \(E(X^2) = \frac{12}{7}\) | B1; M1 A1 | |
| (iii) \(\text{Var}(X) = \frac{12}{7} - \frac{64}{49} = \frac{20}{49}\) | M1 A1 | |
| (iii) \(\text{Var}(2X) = 4\text{Var}(X) = \frac{80}{49}\) or 1.63 | M1 A1 | Total: 10 |
(a) A quantity which can take only certain distinct values, with fixed probabilities | B1 |
(b) | x | 0 | 1 | 2 |
| P(X = x) | $\frac{1}{7}$ | $\frac{4}{7}$ | $\frac{2}{7}$ | | M1 A1 A1 A1 |
(c) (i) $E(X) = \frac{8}{7}$ | B1; M1 A1 |
(ii) $E(X^2) = \frac{12}{7}$ | B1; M1 A1 |
(iii) $\text{Var}(X) = \frac{12}{7} - \frac{64}{49} = \frac{20}{49}$ | M1 A1 |
(iii) $\text{Var}(2X) = 4\text{Var}(X) = \frac{80}{49}$ or 1.63 | M1 A1 | **Total: 10**\begin{enumerate}[label=(\alph*)]
\item Explain briefly what is meant by a discrete random variable. [1 mark]
A family has 3 cats and 4 dogs. Two of the family's animals are to be chosen at random. The random variable $X$ represents the number of dogs chosen.
\item Copy and complete the table to show the probability distribution of $X$:
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 \\
\hline
P$(X = x)$ & & & \\
\hline
\end{tabular}
[4 marks]
\item Calculate \begin{enumerate}[label=(\roman*)]
\item E$(X)$,
\item Var$(X)$,
\item Var$(2X)$.
\end{enumerate}
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q1 [10]}}