The geometric distributionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A machine produces components, each of which is defective with probability $0.08$, independently of all the others. The components are inspected one at a time. Let $X$ be the number of components inspected up to and including the first defective one.

   (a)

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

   [1 mark]

   • A$0.0573$
   • B$0.0623$
   • C$0.7787$
   • D$0.0005$

   (b)

   Find the probability that the first defective component is not found until after the 5th inspection, that is $\mathrm{P}(X>5)$.

   [1 mark]

   • A$0.3409$
   • B$0.6064$
   • C$0.6591$
   • D$0.7164$

   (c)

   Find the smallest number $n$ of inspections for which the probability that the first defective component has been found by the $n$th inspection exceeds $0.5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A driving test candidate passes at each attempt with probability $0.35$, independently of any other attempt. Let $X$ be the number of attempts the candidate makes up to and including the first one that they pass.

   (a)

   Find the probability that the candidate passes at the third attempt.

   [1 mark]

   • A$0.1479$
   • B$0.0796$
   • C$0.2746$
   • D$0.4225$

   (b)

   Find the probability that the candidate passes within three attempts, that is $\mathrm{P}(X\leq3)$.

   [1 mark]

   • A$0.2746$
   • B$0.5775$
   • C$0.1479$
   • D$0.7254$

   (c)

   Find the probability that the candidate needs at least $5$ attempts to pass, and the expected number of attempts needed.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An archer hits the target on each shot with probability $p$, independently of all other shots. Let $X$ be the number of shots up to and including the first hit. It is known that $\mathrm{E}(X)=5$.

   (a)

   Find the value of $p$, and the variance and standard deviation of $X$.

   [3 marks]

   (b)

   Find $\mathrm{P}(X\geq4)$, and find the probability that the first hit comes on an even-numbered shot.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Alice and Ben take turns to roll a die, with Alice rolling first. The winner is the first person to roll a six. Let $X$ be the total number of rolls made up to and including the first six.

   (a)

   (i) State the distribution of $X$ when the die is fair, and find $\mathrm{Var}(X)$.
   (ii) Show that the probability that Alice wins is $\frac{6}{11}$.

   [6 marks]

   (b)

   The die is replaced by a biased die that shows a six with probability $p$, where $0<p<1$. Show that the probability that Alice wins is $\frac{1}{2-p}$, and hence find the value of $p$ for which Alice wins with probability $0.6$.

   [6 marks]

   Total for question 4: 12 marks

End of questions