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AS A-level Statistics · Edexcel 8MA0 · free A-level maths worksheet

Conditional probability

P(A | B) with Venn diagrams, tree diagrams and two-way tables, the addition and multiplication formulae, testing for independence.

12 original questions, 82 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

The events A and B are such that

P(A) = 0.4   P(B) = 0.35   P(A ∪ B) = 0.6

  1. Find P(A ∩ B)
  2. Find P(A | B)
  3. Determine whether or not A and B are independent. Show your working clearly.
4 marks
Show answer
a) 0.15 b) 3/7 (0.429) c) P(A)P(B) = 0.14 ≠ 0.15 = P(A ∩ B), so A and B are not independent
2

The table shows how the 200 students at a sixth form college travel to college.

BusCarWalkTotal
Year 1238223090
Year 13424028110
Total806258200

A student is chosen at random.

  1. Find the probability that the student walks to college.
  2. Given that the student travels by car, find the probability that the student is in Year 13
  3. Find P(bus | Year 12)

The event W is that the student walks and the event T is that the student is in Year 12

  1. Determine whether or not W and T are independent.
5 marks
Show answer
a) 29/100 (0.29) b) 20/31 (0.645) c) 19/45 (0.422) d) P(W)P(T) = 0.29 × 0.45 = 0.1305 ≠ 0.15 = P(W ∩ T), so not independent
3

The events A and B are such that

P(A) = 0.45   P(B | A) = 0.4   P(A′ ∩ B′) = 0.25

  1. Draw a Venn diagram to illustrate the events A and B, giving the probability for each region.
  2. Find P(A | B)
  3. Determine whether or not A and B are independent.
ξAB
6 marks
Show answer
a) A only 0.27, A ∩ B 0.18, B only 0.30, outside 0.25 b) 0.375 c) P(A)P(B) = 0.45 × 0.48 = 0.216 ≠ 0.18 = P(A ∩ B), so not independent
4

When Leila serves in tennis, her first serve is in with probability 0.6. If her first serve is in, she wins the point with probability 0.7

If her first serve is out, she takes a second serve. Her second serve is in with probability 0.9 and, if it is in, she wins the point with probability 0.5. If her second serve is out, she loses the point.

  1. Find the probability that Leila wins a point when she serves.
  2. Given that Leila wins a point, find the probability that her first serve was in.
  3. Given that Leila loses a point, find the probability that both of her serves were out.
6 marks
Show answer
a) 0.6 b) 0.7 c) 0.1
5

The Venn diagram, where p, q and r are probabilities, shows the events A, B and C and their associated probabilities.

  1. State a pair of mutually exclusive events.

Given that P(B) = 0.4 and that the events B and C are independent,

  1. find the value of q
  2. find the value of p and the value of r
  3. Find P(A | B′)
  4. Find P(B ∩ C′ | A ∪ C)
ξABC0.12p0.150.080q0r
8 marks
Show answer
a) A and C b) q = 0.1 c) p = 0.22, r = 0.33 d) 0.2 e) 8/45 (0.178)
6

For the events A and B,

P(A) = 0.3   P(B | A) = 0.5   P(A | B) = 0.25

  1. Find P(B)
  2. Find P(A ∪ B)
  3. Find P(A′ | B′)
6 marks
Show answer
a) 0.6 b) 0.75 c) 0.625
7

Hamza has two bags of beads. Bag X contains 6 red beads and 4 blue beads. Bag Y contains only red beads and blue beads.

Hamza takes a bead at random from bag X and then a bead at random from bag Y. The probability that both beads are blue is 0.12

  1. Find the probability of taking a blue bead from bag Y
  2. Complete the tree diagram.
  3. Find the probability that the two beads are different colours.
  4. Given that the two beads are different colours, find the probability that the bead from bag X was red.
Bag XBag YRedRedBlueBlueRedBlue
7 marks
Show answer
a) 0.3 b) X: red 0.6, blue 0.4; Y: red 0.7, blue 0.3 on both branches c) 0.46 d) 9/23 (0.391)
8

The events A and B are independent, with P(A) > P(B).

P(A ∩ B) = 0.12   P(A ∪ B) = 0.58

  1. Find P(A) and P(B)
  2. Find P(A′ | B)
  3. Prove that, for any two independent events A and B, the events A and B′ are also independent.
7 marks
Show answer
a) P(A) = 0.4, P(B) = 0.3 b) 0.6 c) P(A ∩ B′) = P(A) - P(A ∩ B) = P(A) - P(A)P(B) = P(A)(1 - P(B)) = P(A)P(B′), so A and B′ are independent
9

A leisure centre has 500 members. The table shows the age group of each member and whether or not they live in the town.

Lives in townLives outside town
Junior12030
Adult16090
Senior6040

Of the members, 60% of juniors, 30% of adults and 20% of seniors swim, wherever they live.

A member is chosen at random. The event J is that the member is a junior, S that the member swims and T that the member lives in the town.

  1. Find the probability that the member is an adult who lives outside the town.
  2. Complete the Venn diagram for the events J, S and T, giving the probability for each region.
  3. Find P(S | T)
  4. Find P(J | S)
  5. Find P([S ∪ T]′)
ξJST
9 marks
Show answer
a) 0.18 b) J only 0.024, S only 0.07, T only 0.32, J ∩ S only 0.036, J ∩ T only 0.096, S ∩ T only 0.12, J ∩ S ∩ T 0.144, outside 0.19 c) 33/85 (0.388) d) 18/37 (0.486) e) 0.214
10

A company estimates that 15% of the emails it receives are spam. Its filter labels a spam email as spam with probability 0.9 and labels a genuine email as spam with probability 0.05

  1. Find the probability that a randomly chosen email is labelled as spam.
  2. Given that an email is labelled as spam, find the probability that it is actually spam.
  3. Two emails that have been labelled as spam are chosen at random. Find the probability that exactly one of them is genuine.
  4. The company says that the filter is “right 90% of the time”. Use your answer to part (b) to comment on this statement.
7 marks
Show answer
a) 0.1775 b) 0.761 c) 0.364 d) only about 76% of the emails labelled spam are actually spam, so about 1 in 4 is a genuine email wrongly labelled
11

The events A, B and C are such that B and C are mutually exclusive. The Venn diagram, where x and y are probabilities, shows the probability of each region.

  1. Find the greatest possible value of P(B | A)

Given also that A and C are independent and that A and B are not mutually exclusive,

  1. find the value of x and the value of y
  2. find P(A ∪ B | C′)
  3. Use set notation to write down an event with probability 0.2
ξABC0.150.2yx0.1000.25
8 marks
Show answer
a) 6/11 (0.545) b) x = 0.15, y = 0.15 c) 2/3 d) B ∩ A′
12

A factory makes bike lights on three machines. Machine A makes 50% of the lights, machine B makes 30% and machine C makes 20%.

Of the lights made on machine A, 2% are faulty, and of the lights made on machine C, 5% are faulty. The percentage of lights made on machine B that are faulty is k%.

Given that a randomly chosen faulty light was made on machine B with probability 0.36,

  1. show that k = 3.75
  2. find the probability that a randomly chosen light is faulty.

The lights are packed in boxes of 40, chosen at random from all the lights made.

  1. Find the probability that a box contains more than 2 faulty lights.
  2. A box is found to contain exactly one faulty light. Find the probability that this light was made on machine A.
9 marks
Show answer
a) 0.3d / (0.02 + 0.3d) = 0.36 gives d = 0.0375, so k = 3.75 b) 0.03125 c) 0.129 d) 0.32

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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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