Skip to content

Paper Together: sit a paper in a room with your friends.See how it works

Conditional probability practice questions with answers

Higher · Grade 7 · Non-calculator

A second set of conditional probability questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

Two counters are drawn without replacement. 7 red and 3 blue. What is the probability the second is red given that the first was red?

Show answer

Answer: 2/3

  1. Red remaining: 7 − 1 = 6; total remaining: 10 − 1 = 9.
  2. 6/9 simplifies to 2/3.
  • Method (1): Red remaining: 7 − 1 = 6; total remaining: 10 − 1 = 9.
  • Accuracy (1): 6/9 simplifies to 2/3.

Question 2 Easy

Two counters are drawn without replacement. 5 red and 3 blue. What is the probability the first is red?

Show answer

Answer: 5/8

  1. Initial red: 5; initial total: 8.
  2. P = 5/8.
  • Method (1): Initial red: 5; initial total: 8.
  • Accuracy (1): P = 5/8.

Question 3 Medium

Two counters are drawn without replacement. 7 red and 3 blue. What is the probability both are red?

Show answer

Answer: 7/15

  1. First red: 7/10; second red: 6/9 = 2/3.
  2. 7/10 × 2/3 = 14/30 = 7/15.
  • Method (1): First red: 7/10; second red: 6/9 = 2/3.
  • Accuracy (1): 7/10 × 2/3 = 14/30 = 7/15.

Question 4 Medium

Two counters are drawn without replacement. 6 red and 4 blue. What is the probability the first is red?

Show answer

Answer: 3/5

  1. Initial red: 6; initial total: 10.
  2. 6/10 simplifies to 3/5.
  • Method (1): Initial red: 6; initial total: 10.
  • Accuracy (1): 6/10 simplifies to 3/5.

Question 5 Hard

Two counters are drawn without replacement. 4 red and 6 blue. What is the probability the second is blue given that the first was red?

Show answer

Answer: 2/3

  1. Blue remaining: 6; total remaining: 9.
  2. 6/9 simplifies to 2/3.
  • Method (1): Blue remaining: 6; total remaining: 9.
  • Accuracy (1): 6/9 simplifies to 2/3.

Question 6 Hard

Two counters are drawn without replacement. 8 red and 2 blue. What is the probability the second is red given that the first was red?

Show answer

Answer: 7/9

  1. Red remaining: 8 − 1 = 7; total remaining: 10 − 1 = 9.
  2. P = 7/9.
  • Method (1): Red remaining: 8 − 1 = 7; total remaining: 10 − 1 = 9.
  • Accuracy (1): P = 7/9.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Two counters are drawn without replacement from a bag of 7 red and 5 blue counters. Work out the probability that both are red.

Show answer

Answer: 7/22

  1. Find probability of first red: 7/12.
  2. Find probability of second red without replacement: 6/11.
  3. Multiply branch probabilities: (7/12) × (6/11).
  4. Calculate product: 42/132 = 7/22.
  • Method (1): (7/12) × (6/11)
  • Accuracy (1): 7/22

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

5 red and 3 blue counters; the first red is not replaced. What is the probability the second is red?

  1. 4/7
  2. 5/8
  3. 5/7
  4. 4/8
Show answer

4/7 One red is gone: 4 red remain out of 7 total.

Check 2

Why does the denominator decrease by 1 on the second draw without replacement?

  1. one counter has been removed from the bag and not returned
  2. the counters become smaller
  3. probabilities must always decrease
  4. red counters count as two
Show answer

one counter has been removed from the bag and not returned Removing an item without replacing it reduces the total count by 1.

Check 3

Two counters are drawn without replacement from 5 red and 3 blue. What is P(both are red)?

  1. 5/14
  2. 4/7
  3. 25/64
  4. 9/14
Show answer

5/14 5/8 × 4/7 = 20/56 = 5/14.

Check 4

How do you calculate P(both are red) without replacement?

  1. multiply P(first is red) by P(second is red given first was red)
  2. add the two probabilities
  3. multiply by 2
  4. square the first probability
Show answer

multiply P(first is red) by P(second is red given first was red) Use the multiplication rule: P(A and B) = P(A) × P(B|A).

Check 5

Given the first was red, what is the probability the second is blue from 5 red and 3 blue?

  1. 3/7
  2. 3/8
  3. 4/7
  4. 5/7
Show answer

3/7 All 3 blue counters remain, but total is now 7.

Check 6

Why should conditional probabilities be written as fractions rather than rounded decimals?

  1. fractions give the exact mathematical value without rounding error
  2. decimals are not allowed in GCSE
  3. fractions are always smaller
  4. decimals cannot be multiplied
Show answer

fractions give the exact mathematical value without rounding error Fractions such as 4/7 or 5/14 are exact, whereas 0.571 is an approximation.

Check 7

6 red and 4 blue counters without replacement. What is P(second red | first red)?

  1. 5/9
  2. 6/10
  3. 4/9
  4. 3/5
Show answer

5/9 5 red remaining out of 9 total counters.

Check 8

6 red and 4 blue counters without replacement. What is P(both are red)?

  1. 1/3
  2. 5/9
  3. 9/25
  4. 3/5
Show answer

1/3 6/10 × 5/9 = 30/90 = 1/3.

Check 9

Before any counters are removed, what is P(first is red) from 6 red and 4 blue? Give it in its simplest form.

  1. 3/5
  2. 6/10
  3. 5/9
  4. 1/2
Show answer

3/5 6/10 simplifies to 3/5.

Check 10

4 red and 4 blue counters without replacement. What is P(both are red)?

  1. 3/14
  2. 1/4
  3. 1/2
  4. 3/7
Show answer

3/14 4/8 × 3/7 = 1/2 × 3/7 = 3/14.

Check 11

7 red and 3 blue counters without replacement. What is P(both are red)?

  1. 7/15
  2. 2/3
  3. 49/100
  4. 7/10
Show answer

7/15 7/10 × 6/9 = 7/10 × 2/3 = 14/30 = 7/15.

Check 12

A student writes 0.333 for the probability of both red from 6 red and 4 blue. What is the exact fraction?

  1. 1/3
  2. 3/10
  3. 33/100
  4. 1/9
Show answer

1/3 30/90 cancels to 1/3.

Check 13

4 red and 2 blue counters without replacement. What is P(second red | first red)?

  1. 3/5
  2. 4/6
  3. 2/5
  4. 1/2
Show answer

3/5 3 red remaining out of 5 total counters.

Check 14

8 red and 2 blue counters without replacement. What is P(second red | first red)?

  1. 7/9
  2. 8/10
  3. 2/9
  4. 4/5
Show answer

7/9 7 red remaining out of 9 total counters.

Check 15

What two quantities change when drawing without replacement if the first colour is red?

  1. both the red count and the total count decrease by 1
  2. only the total count changes
  3. only the blue count changes
  4. neither count changes
Show answer

both the red count and the total count decrease by 1 One red is removed, so red decreases by 1 and total decreases by 1.

Keep going with Conditional probability

Practise free

Start free

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Next topicPractise free