| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | One unknown from sum constraint only |
| Difficulty | Easy -1.8 This is a very straightforward probability distribution question requiring only basic recall: (a) uses the fact that probabilities sum to 1 (trivial arithmetic), (b) applies standard E(X) and Var(X) formulas with given values, and (c) involves simple interpretation. No problem-solving or novel insight needed—purely mechanical application of definitions. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 2 | 100 | 1000 |
| \(\mathrm { P } ( X = x )\) | 0.9 | 0.09 | \(p\) | 0.0001 |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) | \(p = 0.0099\) | B1 |
| 1(b) | \(E(X) = (0 \times 0.9 + 2) \times 0.09 + 100 \times 0.0099 + 1000 \times 0.0001\) | M1 |
| \(E(X) = 1.27\) | A1 M1 | FT their \(p\) and "their \(E(X)\)" Allow one slip |
| \(\text{Var}(X) = (0^2 \times 0.9 + 2)^2 \times 0.09 + 100^2 \times 0.0099 + 1000^2 \times 0.0001 - 1.27^2\) | A1 M1 | |
| \(\text{Var}(X) = 197.7(471)\) | A1 | Accept 198 from correct working |
| 1(c)(i) | £1.28 | B1 |
| 1(c)(ii) | Valid explanation. e.g. People may be willing to pay for the excitement of the lottery. The lottery may be raising money for charity. People don't often make decisions based on mathematics. People could win a lot of money. | E1 |
| Total [7] |
1(a) | $p = 0.0099$ | B1 |
1(b) | $E(X) = (0 \times 0.9 + 2) \times 0.09 + 100 \times 0.0099 + 1000 \times 0.0001$ | M1 | FT "their $p$" Allow one slip
| $E(X) = 1.27$ | A1 M1 | FT their $p$ and "their $E(X)$" Allow one slip
| $\text{Var}(X) = (0^2 \times 0.9 + 2)^2 \times 0.09 + 100^2 \times 0.0099 + 1000^2 \times 0.0001 - 1.27^2$ | A1 M1 |
| $\text{Var}(X) = 197.7(471)$ | A1 | Accept 198 from correct working
1(c)(i) | £1.28 | B1 | FT their $E(X)$
1(c)(ii) | Valid explanation. e.g. People may be willing to pay for the excitement of the lottery. The lottery may be raising money for charity. People don't often make decisions based on mathematics. People could win a lot of money. | E1 |
| **Total [7]** |
---\begin{enumerate}
\item The probability distribution for the prize money, $\pounds X$ per ticket, in a local fundraising lottery is shown below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
$x$ & 0 & 2 & 100 & 1000 \\
\hline
$\mathrm { P } ( X = x )$ & 0.9 & 0.09 & $p$ & 0.0001 \\
\hline
\end{tabular}
\end{center}
(a) Calculate the value of $p$.\\
(b) Find $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.\\
(c) (i) What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?\\
(ii) Suggest why, in practice, people would be prepared to pay more than this minimum price.\\
\hfill \mbox{\textit{WJEC Further Unit 2 2022 Q1 [7]}}