Statistics 2 (S2): The Binomial and Poisson distributionsEdexcel International A Level Further Maths: Topic test

1. 1
   A basketball player scores with each free throw with probability $0.7$, independently of other throws. In a practice session she takes $10$ free throws, and $X$ is the number that she scores.

   (a)

   Find $\mathrm{P}(X=8)$.

   [1 mark]

   • A$0.233$
   • B$0.121$
   • C$0.00519$
   • D$0.851$

   (b)

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

   [1 mark]

   • A$7$
   • B$4.9$
   • C$2.1$
   • D$1.45$

   (c)

   Find the probability that she scores at least $8$ of the $10$ free throws.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A telescope records meteors at random at a constant mean rate of $2.5$ per $10$ minutes. The number of meteors recorded in a $10$-minute period is $X$.

   (a)

   Find $\mathrm{P}(X=2)$.

   [1 mark]

   • A$0.0821$
   • B$0.257$
   • C$0.205$
   • D$0.544$

   (b)

   Which distribution models the number of meteors recorded in a $30$-minute period?

   [1 mark]

   • A$\mathrm{Po}(2.5)$
   • B$\mathrm{Po}(0.833)$
   • C$\mathrm{Po}(75)$
   • D$\mathrm{Po}(7.5)$

   (c)

   Find the probability that at least one meteor is recorded in a $10$-minute period.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Each morning, a train on a certain route runs on time with probability $0.85$, independently of other mornings. Over a period of $12$ mornings, $X$ is the number of mornings on which the train runs on time.

   (a)

   Find the probability that the train runs on time on at least $10$ of the $12$ mornings.

   [3 marks]

   (b)

   Three separate periods of $12$ mornings are considered. Find the probability that the train runs on time on at least $10$ mornings in at least two of the three periods.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A small shop is open for business each day. Customers enter at random at a constant mean rate of $12$ per hour. The shop's card reader fails to process a payment with probability $0.01$, independently for each payment.

   (a)

   Find (i) the probability that exactly $2$ customers enter in a $10$-minute period, (ii) the probability that more than $8$ customers enter in a $30$-minute period, (iii) the probability that, in $6$ separate $10$-minute periods, exactly $2$ of the periods have no customers entering.

   [6 marks]

   (b)

   On a busy day the card reader processes $300$ payments. Let $F$ be the number of payments that fail. State the distribution of $F$, and explain why a Poisson distribution is a suitable approximation. Use the Poisson approximation to find $\mathrm{P}(F\ge5)$, and use the exact distribution to find the same probability.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A pottery makes ceramic mugs. Each mug independently has a chip with probability $0.015$. A pallet carries $200$ mugs, and the number of chipped mugs on a pallet is $X$.

   (a)

   Which distribution is the most suitable approximation to the distribution of $X$?

   [1 mark]

   • A$\mathrm{Po}(200)$
   • B$\mathrm{Po}(0.015)$
   • C$\mathrm{Po}(3)$
   • D$\mathrm{Po}(2.955)$

   (b)

   Use a suitable Poisson approximation to find $\mathrm{P}(X\le1)$.

   [1 mark]

   • A$0.199$
   • B$0.0498$
   • C$0.149$
   • D$0.423$

   (c)

   Use the Poisson approximation to find the probability that a pallet has more than $5$ chipped mugs.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A driving instructor knows that each of her learners passes the practical test at the first attempt with probability $0.4$, independently of other learners. She has $15$ learners, and $X$ is the number of them who pass at the first attempt.

   (a)

   Find $\mathrm{P}(X=6)$.

   [1 mark]

   • A$0.610$
   • B$0.0612$
   • C$0.0000413$
   • D$0.207$

   (b)

   Find the standard deviation of $X$.

   [1 mark]

   • A$3.6$
   • B$1.90$
   • C$6$
   • D$2.45$

   (c)

   Find the probability that at least $9$ of her learners pass at the first attempt.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Patients arrive at a hospital emergency department at random at a constant mean rate of $9$ per hour. The number of patients arriving in a $20$-minute period is $X$.

   (a)

   Find the probability that exactly $4$ patients arrive in a $20$-minute period.

   [3 marks]

   (b)

   Find the smallest integer $k$ such that $\mathrm{P}(X>k)<0.01$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A bookbinder finds that each book that she binds independently has a faulty binding with probability $0.004$. A distributor receives consignments of $500$ books, and the number of faulty books in a consignment is $X$.

   (a)

   State the mean and variance of $X$, and explain how these values show that a Poisson distribution is a suitable approximation. Use a suitable Poisson approximation to find the probability that a consignment has at least $4$ faulty books.

   [6 marks]

   (b)

   A consignment is rejected if it contains at least $4$ faulty books. Use $p=0.143$ for the probability that a consignment is rejected. The distributor receives $10$ independent consignments. Find the probability that exactly $2$ are rejected. Find the smallest number $n$ of independent consignments for which the probability that at least one is rejected exceeds $0.9$.

   [6 marks]

   Total for question 8: 12 marks

End of questions