Probability & Statistics 1: A Level Mathematics questions and answers
10 free A Level Mathematics Probability & Statistics 1 questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Representation of data, Permutations & combinations, Probability & the binomial distribution, Discrete random variables, The normal distribution. Work through the questions below, then generate a full Probability & Statistics 1 paper with a fresh set every time.
The lengths of a type of fish are normally distributed with mean μ cm and standard deviation σ cm. 27.43% of the fish are shorter than 51.4 cm and 18.41% are longer than 67.9 cm.
(a)
Find the values of μ and σ.
[5]
(b)
35% of fish in another lake are diseased. A random sample of 50 fish is taken. Use a suitable approximation to find the probability that at least 21 fish are diseased.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
μ = 58, σ = 11
z = -0.600 or 0.900 seen B1 51.4 − μσ = -0.600 M1 67.9 − μσ = 0.900 M1 solve simultaneously M1 μ = 58, σ = 11 A1
Bag X contains 5 red and 2 blue discs. Bag Y contains 6 red and 4 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.
valid comparison of probabilities M1 correct conclusion A1
2
Question 3Structured · 5 marksA* challenge
A game is played repeatedly until a player wins for the first time. The probability of winning any game is 16, 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
Part
Answer and marking points
Marks
(a)
0.0670
geometric distribution: qk−1p M1 0.0670 A1
2
(b)
0.4019
q5M1 0.4019 A1
2
(c)
6
6 B1
1
Question 4Structured · 7 marksA* challenge
The times, x minutes, taken by 23 students to complete a puzzle are summarised by Σx = 1587 and Σx2 = 110630. Another 15 students had mean time 61 minutes and standard deviation 10 minutes.
(a)
Find the mean and standard deviation of the times of the first group.
[2]
(b)
Find the mean and standard deviation of the times of all 38 students.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
69, 7
mean 69 B1 √(110630 ÷ 23 − 692) = 7 B1
2
(b)
65.8, 9.19
Σx for second group = 915 B1 Σx2 for second group = 57315 M1 combined mean 65.8 A1 combined variance formula M1 s.d. 9.19 A1
5
Question 5Structured · 3 marks
The lengths of rods are normally distributed with mean 50 cm. 20% of rods are longer than 65 cm. Find the standard deviation.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
17.8 cm
z = 0.8416 B1 65 − 50σ = 0.8416 M1 17.8 A1
3
Question 6Structured · 5 marks
On average 10% of seeds of a type fail to germinate. 8 seeds are planted.
(a)
Find the probability that exactly 2 seeds fail to germinate.
[2]
(b)
Find the probability that fewer than 3 seeds fail to germinate.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.149
8C2 × 0.12 × 0.96M1 0.149 A1
2
(b)
0.962
P(0) + P(1) + P(2) M1 Correct terms A1 0.962 A1
3
Question 7Structured · 5 marks
A biased coin has probability 0.2 of showing heads. It is thrown 4 times. X is the number of heads.
(a)
Draw up the probability distribution table for X.
[3]
(b)
Find E(X) and Var(X).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
binomial probabilities
binomial form M1 at least two correct A1 all correct A1
3
(b)
0.8; 0.64
E(X) = 0.8 B1 Var(X) = 0.64 B1
2
Question 8Structured · 2 marks
Find the number of different arrangements of the 7 letters in the word SUCCESS.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
420
7! ÷ (3! × 2!) M1 420 A1
2
Question 9Structured · 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
Part
Answer and marking points
Marks
(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 10Structured · 5 marks
The random variable X has the probability distribution shown.
What is in A Level Mathematics Probability & Statistics 1?
In the 9709 syllabus, Probability & Statistics 1 covers Representation of data, Permutations & combinations, Probability & the binomial distribution, Discrete random variables, The normal distribution.
Where can I get Probability & Statistics 1 questions with mark schemes?
This page has 10 Probability & Statistics 1 questions with answers. For more, the free generator on mentorriyaz.in builds a new Probability & Statistics 1 paper with a full mark scheme each time you click.
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Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in