Further Statistics 1: Geometric and negative binomial distributionsEdexcel A-Level Further Maths: Topic test

1. 1
   A botanist plants seeds one at a time. Each seed germinates with probability $0.6$, independently of all the others. Let $X$ be the number of seeds planted up to and including the first one that germinates.

   (a)

   What is $\mathrm{P}(X=3)$?

   [1 mark]

   • A$0.096$
   • B$0.144$
   • C$0.216$
   • D$0.064$

   (b)

   What is $\mathrm{Var}(X)$?

   [1 mark]

   • A$\frac{2}{3}$
   • B$\frac{10}{9}$
   • C$\frac{5}{3}$
   • D$\frac{25}{9}$

   (c)

   Find the probability that more than $4$ seeds have to be planted before one germinates.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A quality inspector tests batteries one at a time. Each battery is faulty with probability $0.2$, independently of all the others. Let $Y$ be the number of batteries tested up to and including the third faulty battery.

   (a)

   What is $\mathrm{P}(Y=5)$, to 3 significant figures?

   [1 mark]

   • A$0.0512$
   • B$0.00512$
   • C$0.0307$
   • D$0.0205$

   (b)

   What is $\mathrm{E}(Y)$?

   [1 mark]

   • A$0.6$
   • B$5$
   • C$12$
   • D$15$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A darts player throws darts at the bull's-eye. Each throw hits the bull's-eye with probability $p$, independently of all other throws. Let $X$ be the number of throws up to and including her first hit. It is given that $\mathrm{E}(X)=8$.

   (a)

   Find the value of $p$ and hence the probability that her first hit is on her third throw.

   [3 marks]

   (b)

   Find $\mathrm{Var}(X)$ and the probability that she needs more than $10$ throws to score her first hit.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A laboratory screens blood samples one at a time. Each sample tests positive with probability $0.15$, independently of all other samples. Let $X$ be the number of samples screened up to and including the first positive sample, and let $Y$ be the number screened up to and including the third positive sample.

   (a)

   Find (i) $\mathrm{P}(X=5)$, (ii) $\mathrm{P}(Y=5)$, (iii) $\mathrm{Var}(Y)$. Give each answer to 3 significant figures. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   The first positive sample is the $4$th sample screened. Let $W$ be the number of further samples screened up to and including the third positive sample overall. (i) Explain why $W$ has a negative binomial distribution with $r=2$ and $p=0.15$. (ii) Hence find the probability that $Y=9$ given that the first positive sample is the $4$th sample. Marks: (i) 3, (ii) 3.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   On any morning, a commuter's bus is late with probability $0.2$, independently of other mornings. Let $X$ be the number of mornings, starting from tomorrow, up to and including the first morning on which the bus is late.

   (a)

   Which is the distribution of $X$?

   [1 mark]

   • A$\mathrm{Geo}(0.8)$
   • B$\mathrm{Geo}(0.2)$
   • C$\mathrm{B}(5,0.2)$
   • D$\mathrm{NB}(2,0.2)$

   (b)

   What is $\mathrm{P}(X\leq3)$?

   [1 mark]

   • A$0.488$
   • B$0.512$
   • C$0.2$
   • D$0.128$

   (c)

   Find the standard deviation of $X$, to 3 significant figures.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A tennis player attempts first serves. Each first serve goes in with probability $0.6$, independently of all other first serves. Let $Z$ be the number of first serves she attempts up to and including her fourth successful one.

   (a)

   What is $\mathrm{P}(Z=4)$?

   [1 mark]

   • A$0.3456$
   • B$0.0256$
   • C$0.2160$
   • D$0.1296$

   (b)

   What is $\mathrm{E}(Z)$?

   [1 mark]

   • A$2.4$
   • B$\frac53$
   • C$\frac{20}{3}$
   • D$\frac83$

   (c)

   Find the probability that she needs exactly $6$ first serves to achieve her fourth success.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   An angler casts a line repeatedly. On each cast, independently of all other casts, she catches a fish with probability $p$, where $p>0.5$. Let $X$ be the number of casts up to and including her first catch. It is given that $\mathrm{P}(X=2)=0.21$.

   (a)

   Find the value of $p$.

   [3 marks]

   (b)

   Find $\mathrm{E}(X)$ and the probability that her first catch is on an even-numbered cast.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A scientist repeats an experiment until it has succeeded twice. Each experiment succeeds with probability $0.4$, independently of the others. Let $X$ be the total number of experiments performed.

   (a)

   (i) State the distribution of $X$ and find $\mathrm{Var}(X)$. (ii) Find $\mathrm{P}(X=4)$. (iii) Find $\mathrm{P}(X\leq3)$. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   The scientist can afford at most $6$ experiments. (i) Find the probability that she obtains her two successes within $6$ experiments, by considering the number of successes that would occur in $6$ experiments. (ii) Find $\mathrm{P}(X=6)$. Marks: (i) 4, (ii) 2.

   [6 marks]

   Total for question 8: 12 marks

End of questions