A Level · Mathematics 9709 · Probability & Statistics 1

Probability & the binomial distribution: A Level Mathematics questions with answers

Free Probability & the binomial distribution questions for A Level Mathematics (9709), part of Probability & Statistics 1. Each question has the answer and the marking points an examiner would look for.

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Probability & the binomial distribution practice questions

Question 1Structured · 5 marksA* challenge

A game is played repeatedly until a player wins for the first time. The probability of winning any game is 0.25, independently.

(a)

Find the probability that the first win is on the 4th game.

[2]
(b)

Find the probability that more than 8 games are needed.

[2]
(c)

State the expected number of games needed for the first win.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.1055
geometric distribution: qk−1p M1
0.1055 A1
2
(b)0.1001
q8 M1
0.1001 A1
2
(c)4
4 B1
1
Question 2Structured · 6 marksA* challenge

On average 12% of the bolts made by a machine are faulty.

(a)

Find the probability that, in a box of 12 bolts, more than 2 are faulty.

[3]
(b)

Find the least number of bolts that must be checked for the probability of finding at least one faulty bolt to be greater than 0.95.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.1667
1 − P(X ≤ 2) M1
binomial terms correct A1
0.1667 A1
3
(b)24
1 − 0.88n > 0.95 M1
logarithms or trial M1
24 A1
3
Question 3Structured · 5 marks

On average 25% of seeds of a type fail to germinate. 12 seeds are planted.

(a)

Find the probability that exactly 3 seeds fail to germinate.

[2]
(b)

Find the probability that fewer than 3 seeds fail to germinate.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.258
12C3 × 0.253 × 0.759 M1
0.258 A1
2
(b)0.391
P(0) + P(1) + P(2) M1
Correct terms A1
0.391 A1
3
Question 4Structured · 4 marks

Events A and B satisfy P(A) = 0.3, P(B) = 0.2 and P(A ∩ B) = 0.06.

(a)

Show that A and B are independent.

[1]
(b)

Find P(A ∪ B) and P(A | B′).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)P(A)P(B) = P(A ∩ B)
0.3 × 0.2 = 0.06 B1
1
(b)0.44; 0.3
P(A ∪ B) = 0.44 B1
conditional formula M1
0.3 A1
3
Question 5Structured · 7 marksA* challenge

Bag X contains 4 red and 5 blue discs. Bag Y contains 6 red and 3 blue discs. A biased coin, with P(head) = 0.4, is thrown. If it shows a head, a disc is taken at random from bag X; otherwise a disc is taken at random from bag Y.

(a)

Find the probability that the disc is red.

[2]
(b)

Given that the disc is red, find the probability that it was taken from bag X.

[3]
(c)

Determine whether the events "the coin shows a head" and "the disc is red" are independent. Justify your answer.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.5778
sum of two products M1
0.5778 A1
2
(b)0.3077
P(X and red) ÷ P(red) M1
0.1778 ÷ 0.5778 A1
0.3077 A1
3
(c)not independent: P(red | head) = 0.4444 ≠ P(red) = 0.5778
valid comparison of probabilities M1
correct conclusion A1
2
Question 6Structured · 5 marksA* challenge

A game is played repeatedly until a player wins for the first time. The probability of winning any game is 0.25, independently.

(a)

Find the probability that the first win is on the 6th game.

[2]
(b)

Find the probability that more than 5 games are needed.

[2]
(c)

State the expected number of games needed for the first win.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0.0593
geometric distribution: qk−1p M1
0.0593 A1
2
(b)0.2373
q5 M1
0.2373 A1
2
(c)4
4 B1
1

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