| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | Specimen |
| Marks | 11 |
| Topic | Discrete Probability Distributions |
| Type | Construct probability distribution from scenario |
| Difficulty | Moderate -0.8 This is a straightforward probability distribution question requiring basic understanding of net gain/loss, expected value calculation (formula given), and binomial probability. The multi-part structure and 10-mark allocation are typical, but each step involves standard textbook techniques with no novel insight or complex reasoning required. |
| Spec | 2.04a Discrete probability distributions2.04b Binomial distribution: as model B(n,p)2.04e Normal distribution: as model N(mu, sigma^2)2.04h Select appropriate distribution |
**Question 5**
(i) Table shows $(-1, 0.7)$ [B1]
$(0, 0.25)$ and $(9, 0.05)$ [B1]
(ii) Use $E(X)$ formula [M1]
Obtain $-0.25$ AG [A1]
Use $E(X^2)$ formula [M1]
Obtain $4.6875$ (or $4.69$) o.e. [A1]
(iii) Use $10 + 10E(X)$ [M1]
Obtain $10 + 10(-0.25) = 7.5$ [A1]
(iv) P(Must win at least one game) [M1]
States $(0.25)^{10}$ [B1]
Obtain $1 - (0.95)^{10} + (0.25)^{10} = 0.401$ [A1]
**Total: 11 marks**5 James plays an arcade game. Each time he plays, he puts a $\pounds 1$ 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 $\pounds 1$ coin, James wins the game with a probability of 0.05 and the machine pays out ten $\pounds 1$ coins.
The outcomes can be modelled by a random variable $X$ representing the number of $\pounds 1$ coins gained at the end of a game.\\
(i) Construct a probability distribution table for $X$.\\
(ii) Show that $\mathrm { E } ( X ) = - 0.25$ and find $\operatorname { Var } ( X )$.
James starts off with $10 \pounds 1$ coins and decides to play exactly 10 games.\\
(iii) Find the expected number of $\pounds 1$ coins that James will have at the end of his 10 games.\\
(iv) Find the probability that after his 10 games James will have at least $10 \pounds 1$ coins left.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q5 [11]}}