| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Probability distribution from formula |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring standard calculations. Part (a) involves basic probability setup (finding probabilities that sum to 1), part (b) is routine expectation calculation with a given answer to verify, part (c) uses the linear property E(aX+b), and part (d) is standard variance calculation. All techniques are direct applications of formulas with no problem-solving insight required, making it slightly easier than average. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
| Answer | Marks | Guidance |
|---|---|---|
| x | 1 | 2 |
| \(P(x)\) | \(\frac{1}{8}\) | \(\frac{1}{8}\) |
| \(\sum xP(x) = \frac{1}{8}(1 + 2 + 3 + 4 + 5 + 18) = \frac{33}{8}\) | M2 A1 | |
| \((4 \times \frac{33}{8}) - 1 = \frac{31}{2}\) | M1 A1 | |
| \(E(X^2) = \sum x^2P(x) = \frac{1}{8}(1 + 4 + 9 + 16 + 25 + 108) = \frac{163}{8}\) | M1 A1 | |
| \(\text{Var}(X) = \frac{163}{8} - (\frac{33}{8})^2 = \frac{215}{64}\) or 3.36 | M1 A1 | (13) |
| | x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| | $P(x)$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | $\frac{1}{8}$ | | M2 A2 | |
| $\sum xP(x) = \frac{1}{8}(1 + 2 + 3 + 4 + 5 + 18) = \frac{33}{8}$ | M2 A1 | |
| $(4 \times \frac{33}{8}) - 1 = \frac{31}{2}$ | M1 A1 | |
| $E(X^2) = \sum x^2P(x) = \frac{1}{8}(1 + 4 + 9 + 16 + 25 + 108) = \frac{163}{8}$ | M1 A1 | |
| $\text{Var}(X) = \frac{163}{8} - (\frac{33}{8})^2 = \frac{215}{64}$ or 3.36 | M1 A1 | (13) |A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.
\begin{enumerate}[label=(\alph*)]
\item Write down the probability distribution for the random variable, $X$, the score on a single throw of the die. [4]
\item Show that E$(X) = \frac{33}{8}$. [3]
\item Find E$(4X - 1)$. [2]
\item Find Var$(X)$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q4 [13]}}