1. 1
   The discrete random variable $X$ has probability distribution $P(X = x) = kx$ for $x \in \{1, 2, 3, 4\}$, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac{1}{4}$
   • B$\frac{1}{10}$
   • C$\frac{1}{24}$
   • D$10$

   (b)

   Find $E(X)$.

   [1 mark]

   • A$2.5$
   • B$0.75$
   • C$30$
   • D$3$

   (c)

   Find the probability that $X$ is greater than $E(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of goals, $G$, scored by a hockey team in a match is modelled by a discrete random variable with $P(G = 0) = 0.25$, $P(G = 1) = 0.35$, $P(G = 2) = p$, $P(G = 3) = 0.12$ and $P(G = 4) = 0.03$. The team never scores more than 4 goals.

   (a)

   Find the value of $p$.

   [1 mark]

   • A$0.25$
   • B$0.75$
   • C$0.2$
   • D$0.35$

   (b)

   Find the expected number of goals scored in a match.

   [1 mark]

   • A$2$
   • B$0.83$
   • C$1.33$
   • D$0.266$

   (c)

   The team plays 30 matches in a season. Find the expected number of matches in which it scores at least 2 goals.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The discrete random variable $Y$ has probability distribution $P(Y = y) = \frac{2y + 1}{k}$ for $y \in \{0, 1, 2, 3\}$, where $k$ is a constant.

   (a)

   Show that $k = 16$, and hence find $P(Y = 2)$.

   [3 marks]

   (b)

   Find $E(Y)$. Two independent observations of $Y$ are made. Find the probability that their sum is 5.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   At a school fair, a player pays $\$4$ to roll a fair six-sided dice once. If the dice shows 6, the player receives $\$12$. If it shows 4 or 5, the player receives $\$3$. Otherwise the player receives nothing. Let $W$ be the player's gain in dollars (the amount received minus the $\$4$ paid).

   (a)

   (i) Write down the probability distribution of $W$.
   (ii) Find $E(W)$.
   (iii) State, with a reason, whether the game is fair.
   (iv) The game is played 200 times during the fair. Find the organiser's expected profit.

   [6 marks]

   (b)

   (i) The organiser keeps the $\$4$ cost and the $\$3$ prize for a 4 or 5, but changes the prize for a 6 to $\$x$ so that the game is fair. Show that $x = 18$.
   (ii) Alternatively, the organiser keeps the original prizes and changes the cost of a game to $\$c$ so that the game is fair. Find $c$.
   (iii) Suggest why the organiser would not want the game to be fair.

   [6 marks]

   Total for question 4: 12 marks

End of questions