A Level Mathematics: Probability & Statistics 1 Practice Questions (Cambridge 9709 Paper 5)

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — Cambridge International holds copyright in its own papers. Use these alongside the official past papers available from your board.

Each question practises a skill tested in the June 2025 Paper 5 series. After each answer there is a tip and the real question to try next.

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Questions

1. Two coins are thrown together. The first is biased so that P(head) = 1/3; the second is fair. X is the number of heads obtained. Find the probability distribution of X. [3]

2. 60% of the residents of a town own a car. A random sample of 150 residents is chosen. Use a suitable approximation to find the probability that more than 100 of them own a car. [5]

3. A bag contains 5 green marbles and 10 yellow marbles. A marble is taken at random. If it is green it is put back; if it is yellow it is not. A second marble is then taken at random. (a) Show that the probability that the second marble is yellow is 41/63. (b) Find the probability that the first marble was green, given that the second marble is yellow. [4]

4. At a junction, 20% of vehicles turn left, independently of each other. (a) Find the probability that the first vehicle to turn left is the 5th vehicle. (b) Find the probability that the first vehicle to turn left comes before the 6th vehicle. [4]

5. The seven letters of the word BANANAS are arranged in a line. (a) How many different arrangements are there? (b) How many arrangements have no two As next to each other? [5]

6. The wingspans of a species of bird are normally distributed with mean 50 cm and standard deviation 4 cm. In a sample of 200 of these birds, how many would you expect to have a wingspan between 46 cm and 55 cm? [4]

7. The times taken to complete a puzzle are normally distributed with mean μ seconds and standard deviation σ seconds. 10% of people take more than 80 seconds and 25% take less than 60 seconds. Find μ and σ. [5]

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Answers

1. P(X = 0) = 2/3 × 1/2 = 1/3 [1]. P(X = 2) = 1/3 × 1/2 = 1/6 [1]. P(X = 1) = 1 − 1/3 − 1/6 = 1/2 [1].

Tip: check that your probabilities add up to 1.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 1.

2. X ~ B(150, 0.6), approximated by N(90, 36), since np = 90 and npq = 36 are both large [1] [1]. “More than 100” means X ≥ 101, so use 100.5 (continuity correction) [1]. z = (100.5 − 90) ÷ 6 = 1.75 [1]. P(X > 100) ≈ 1 − Φ(1.75) = 1 − 0.9599 = 0.0401 [1].

Tip: write the inequality in whole numbers first, then apply the continuity correction.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 2.

3. (a) P(G then Y) = 5/15 × 10/15 = 2/9 [1]. P(Y then Y) = 10/15 × 9/14 = 3/7. Total = 14/63 + 27/63 = 41/63 [1]. (b) P(first G | second Y) = (2/9) ÷ (41/63) [1] = (14/63) ÷ (41/63) = 14/41 [1].

Tip: the second draw depends on the first, so draw a tree diagram with different second-branch probabilities.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 3.

4. (a) Four vehicles not turning left, then one that does: 0.8⁴ × 0.2 [1] = 0.0819 [1]. (b) “Before the 6th” means within the first 5: 1 − 0.8⁵ [1] = 0.672 [1].

Tip: the probability that the first success happens within n trials is 1 − qⁿ.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 4.

5. (a) 7! ÷ (3! × 2!) [1] = 420 [1]. (b) Arrange B, N, N, S: 4! ÷ 2! = 12 ways [1]. There are 5 gaps for the 3 As: ⁵C₃ = 10 [1]. Total = 12 × 10 = 120 [1].

Tip: for “not together”, arrange the other letters first, then place the repeated letters in the gaps.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 6.

6. z-values: (46 − 50) ÷ 4 = −1 and (55 − 50) ÷ 4 = 1.25 [1]. P = Φ(1.25) − Φ(−1) = 0.8944 − 0.1587 [1] = 0.7357 [1]. Expected number = 200 × 0.7357 ≈ 147 [1].

Tip: standardise both limits, and give the expected number as a whole number at the end.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 7(a).

7. (80 − μ) ÷ σ = 1.282 [1] and (60 − μ) ÷ σ = −0.674 [1]. Subtracting: 20 = 1.956σ [1], so σ = 10.2 [1] and μ = 60 + 0.674 × 10.22 = 66.9 [1].

Tip: use the z-value for the area to the left of the point, and give it a negative sign when that area is below 0.5.

Try the real question next: Cambridge International AS & A Level Mathematics 9709, June 2025, Paper 52, Question 7(b).

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Where marks are usually lost

• No continuity correction, or one applied in the wrong direction.
• Conditional probability found as P(A and B) without dividing by P(B).
• Repeated letters not divided out in arrangements.
• z-values given the wrong sign.