4.7 Discrete random variables and expected valueIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The discrete random variable $X$ has the probability distribution $P(X=1)=0.1$, $P(X=2)=0.3$, $P(X=3)=0.4$ and $P(X=4)=k$.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$0.8$
   • B$0.1$
   • C$0.2$
   • D$0.3$

   (b)

   Find $\mathrm{E}(X)$.

   [1 mark]

   • A$2.7$
   • B$2.5$
   • C$0.25$
   • D$5.1$

   (c)

   Find $P(X\geq3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The discrete random variable $X$ has probability distribution $P(X=x)=\frac{x+4}{18}$ for $x\in\{1,2,3\}$.

   (a)

   Find $P(X=2)$.

   [1 mark]

   • A$\frac{5}{18}$
   • B$\frac{7}{18}$
   • C$\frac16$
   • D$\frac13$

   (b)

   Find $\mathrm{E}(X)$.

   [1 mark]

   • A$2$
   • B$\frac{19}{9}$
   • C$38$
   • D$1$

   (c)

   Find $P(X\geq2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   At a fairground, a player pays 4 AED to spin a wheel. The prize is 0 AED with probability 0.5, 6 AED with probability 0.3 and 10 AED with probability 0.2. Let $X$ be the prize in AED.

   (a)

   Find $\mathrm{E}(X)$.

   [3 marks]

   (b)

   (i) Find the expected gain of a player on one game, after paying to play.
   (ii) State, with a reason, whether the game is fair.
   (iii) The game is played 500 times. Find the organiser's expected profit.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two fair four-sided dice, each with faces numbered 1 to 4, are rolled together. Let $X$ be the larger of the two numbers shown, with $X$ equal to that number if both are the same.

   (a)

   (i) Show that $P(X=3)=\frac{5}{16}$.
   (ii) Given that $P(X=1)=\frac{1}{16}$, find $P(X=2)$ and $P(X=4)$.
   (iii) Find $\mathrm{E}(X)$.

   [6 marks]

   (b)

   A game is played as follows. The player pays 3 AED to roll the two dice and receives $X$ AED back.
   (i) Find the expected gain per game for the player.
   (ii) The player plays 200 times. Find the expected total gain.
   (iii) Comment on whether the game is fair.

   [6 marks]

   Total for question 4: 12 marks

End of questions