A Level · Mathematics 9709 · Probability & Statistics 2

The Poisson distribution: A Level Mathematics questions with answers

Free The Poisson distribution questions for A Level Mathematics (9709), part of Probability & Statistics 2. Each question has the answer and the marking points an examiner would look for.

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The Poisson distribution practice questions

Question 1Structured · 10 marksA* challenge

Calls arrive at a switchboard at random at an average rate of 1.8 per minute.

(a)

Find the probability that at least 3 calls arrive in a 2-minute period.

[3]
(b)

Find the probability that exactly 2 calls arrive in the first minute and exactly 2 in the second minute, given that exactly 4 arrive in the two minutes.

[3]
(c)

1.2% of calls are wrong numbers. Use a suitable approximation to find the probability that, out of 200 calls, fewer than 3 are wrong numbers. Justify your approximation.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.6973
λ = 3.6 B1
1 − P(X ≤ 2) M1
0.6973 A1
3
(b)0.3750
P(2)×P(2) ÷ P(4 in 2 minutes) M1
correct values A1
0.3750 A1
3
(c)0.5697 (Po(2.4): n > 50 and np = 2.4 < 5)
Po(2.4) B1
n large, p small (np < 5) B1
P(0) + P(1) + P(2) M1
0.5697 A1
4
Question 2Structured · 6 marksA* challenge

The numbers of emails received by Sam and by Jo in an hour have independent distributions Po(2) and Po(1.5).

(a)

Find the probability that together they receive more than 3 emails in an hour.

[3]
(b)

Given that they receive 4 emails in total in an hour, find the probability that Sam receives exactly 3 of them.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.4634
Po(3.5) B1
1 − P(X ≤ 3) M1
0.4634 A1
3
(b)0.3199
P(S = 3)P(J = 1) ÷ P(total 4) M1
correct values or binomial with p = 0.571 A1
0.3199 A1
3
Question 3Structured · 5 marks

The number of calls to a helpline in a 5-minute period has the distribution Po(4.5).

(a)

Find the probability of exactly 4 calls in a 5-minute period.

[2]
(b)

Find the probability of at least 2 calls in a 10-minute period.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.19
e−4.5 × 4.54 ÷ 4! M1
0.19 A1
2
(b)0.999
λ = 9 B1
1 − P(0) − P(1) M1
0.999 A1
3
Question 4Structured · 8 marks

Calls to a helpline arrive at random at an average rate of 3 per 10 minutes.

(a)

Find the probability of at least 2 calls in 10 minutes.

[2]
(b)

Find the probability of exactly 3 calls in 30 minutes.

[2]
(c)

Use a suitable approximation to find the probability of more than 50 calls in 3 hours.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.801
1 − P(0) − P(1) M1
0.801 A1
2
(b)0.015
λ = 9 M1
0.015 A1
2
(c)normal approximation
N(54, 54) B1
continuity correction 50.5 M1
standardise M1
probability A1
4
Question 5Structured · 10 marksA* challenge

Calls arrive at a switchboard at random at an average rate of 2.4 per minute.

(a)

Find the probability that at least 3 calls arrive in a 2-minute period.

[3]
(b)

Find the probability that exactly 2 calls arrive in the first minute and exactly 2 in the second minute, given that exactly 4 arrive in the two minutes.

[3]
(c)

1.2% of calls are wrong numbers. Use a suitable approximation to find the probability that, out of 150 calls, fewer than 3 are wrong numbers. Justify your approximation.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.8575
λ = 4.8 B1
1 − P(X ≤ 2) M1
0.8575 A1
3
(b)0.3750
P(2)×P(2) ÷ P(4 in 2 minutes) M1
correct values A1
0.3750 A1
3
(c)0.7306 (Po(1.8): n > 50 and np = 1.8 < 5)
Po(1.8) B1
n large, p small (np < 5) B1
P(0) + P(1) + P(2) M1
0.7306 A1
4
Question 6Structured · 6 marksA* challenge

The numbers of emails received by Sam and by Jo in an hour have independent distributions Po(1.5) and Po(1).

(a)

Find the probability that together they receive more than 2 emails in an hour.

[3]
(b)

Given that they receive 4 emails in total in an hour, find the probability that Sam receives exactly 3 of them.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.4562
Po(2.5) B1
1 − P(X ≤ 2) M1
0.4562 A1
3
(b)0.3456
P(S = 3)P(J = 1) ÷ P(total 4) M1
correct values or binomial with p = 0.6 A1
0.3456 A1
3

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