| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2019 |
| Session | Specimen |
| Marks | 11 |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.3 This is a straightforward probability distribution question requiring basic expectation/variance calculations and binomial probability. Parts (a)-(c) involve routine application of formulas with no conceptual challenges. Part (d) requires recognizing that 'at least 10 coins' means at most 0 net losses, which needs some thought but is still standard probability reasoning. The calculations are mechanical once the setup is understood, making this slightly easier than average. |
| Spec | 2.04a Discrete probability distributions5.02b Expectation and variance: discrete random variables |
James plays an arcade game. Each time he plays, he puts a £1 coin in the slot to start the game. The possible outcomes of each game are as follows:
James loses the game with a probability of 0.7 and the machine pays out nothing,
James draws the game with a probability of 0.25 and the machine pays out a £1 coin,
James wins the game with a probability of 0.05 and the machine pays out ten £1 coins.
The outcomes can be modelled by a random variable $X$ representing the number of £1 coins gained at the end of a game.
\begin{enumerate}[label=(\alph*)]
\item Construct a probability distribution table for $X$. [2]
\item Show that E($X$) = $-0.25$ and find Var($X$). [4]
\end{enumerate}
James starts off with 10 £1 coins and decides to play exactly 10 games.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
\item Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2019 Q5 [11]}}