The negative binomial distributionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A shooter hits the target with probability $0.3$ on each shot, independently of all other shots. Let $X$ be the number of shots needed to score $3$ hits, so that the third hit occurs on shot $X$.

   (a)

   Find $\mathrm{P}(X=5)$.

   [1 mark]

   • A$0.1323$
   • B$0.0132$
   • C$0.1852$
   • D$0.0794$

   (b)

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

   [1 mark]

   • A$\frac{70}{3}$
   • B$7$
   • C$\frac{70}{9}$
   • D$10$

   (c)

   Find the probability that the third hit is scored on or before the fourth shot.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A fair six-sided die is rolled repeatedly. Let $X$ be the number of rolls up to and including the roll on which the second six appears.

   (a)

   Find $\mathrm{P}(X=4)$.

   [1 mark]

   • A$\frac{25}{1296}$
   • B$\frac{25}{432}$
   • C$\frac{25}{216}$
   • D$\frac{125}{432}$

   (b)

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

   [1 mark]

   • A$12$
   • B$10$
   • C$60$
   • D$30$

   (c)

   Find the probability that more than $3$ rolls are needed to obtain the second six.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A telesales agent makes a sale on each call with probability $0.25$, independently of all other calls. Let $X$ be the number of calls made up to and including the call on which the fourth sale is made.

   (a)

   State the distribution of $X$, and find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

   [3 marks]

   (b)

   Find the probability that the agent makes the fourth sale within $5$ calls.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Rovers and United play a series of games in which there are no draws. Rovers win each game with probability $0.6$, independently of the other games. The series ends as soon as one team has won $3$ games. Let $X$ be the number of games Rovers would need to win $3$ games if the games were to continue regardless.

   (a)

   (i) State the distribution of $X$ and one assumption needed for it to be a suitable model.
   (ii) Hence find the probability that Rovers win the series.

   [6 marks]

   (b)

   Let $Y$ be the number of games United would need to win $3$ games if the games were to continue regardless.
   (i) Find $\mathrm{P}(Y=5)$.
   (ii) Hence show that the probability that the series lasts exactly $5$ games is $0.3456$.

   [6 marks]

   Total for question 4: 12 marks

End of questions