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MathidiaMathematics Academy
Combined probability
Lesson 68 of 75
Standard practice

Without replacement

Objective. Update probabilities after an outcome is removed.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Without replacement means an item removed from a set during a probability experiment is not put back, so the total number of items and possibly the number of favourable outcomes change for the next event. This changes each subsequent probability compared with the first draw, requiring a tree diagram or careful fraction tracking across each stage. For example, drawing two red balls in a row from a bag of 5 red and 3 blue balls gives the first probability as 5/8, but the second probability becomes 4/7, since one red ball and one total ball have both been removed. Multiplying these adjusted fractions together gives the probability of both events occurring in sequence.

Worked example

A bag contains 6 green counters and 4 yellow counters. Two counters are drawn without replacement. Find the probability both are green.

6 … 4
Illustrated card showing 6 … 4
  1. 1Find the probability the first counter is green: 6/10.
  2. 2After removing one green counter, 5 green and 9 total counters remain.
  3. 3Find the probability the second counter is also green: 5/9.
  4. 4Multiply the two probabilities: 6/10 × 5/9 = 30/90 = 1/3.

Answer: 1/3

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Practice

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Common mistake. Pupils often keep the same probability for the second draw as the first, forgetting that both the numerator and denominator change once an item is removed without replacement. Always update both numbers in the fraction after each draw without replacement.

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