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Mathematics · Topic

Probability, with and without replacement

Without replacement the second denominator changes. That single detail is where the marks live.

2 original questions in the bank, worth 4 marks · above the two-per-topic floor

Drill this topic All probability, with and without replacement questions

The questions, with the examiner reasoning

Find3 marksIGCSE

A bag contains 5 red counters and 3 blue counters. Two counters are taken at random without replacement. Find the probability that both counters are red, giving your answer as a fraction in its lowest terms.

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Answer: 5/14

  1. First counter red: 5/8
  2. Second counter red, given the first was red: 4/7
  3. 5/8 × 4/7 = 20/56 = 5/14

Examiner technique: Without replacement changes the second fraction on both the top and the bottom. Write the second fraction separately — it is usually a mark of its own.

Find1 markBoth boards

A fair six-sided dice is rolled twice. What is the probability of rolling a six both times?

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Answer: 1/36

  1. Independent events multiply
  2. P(6) × P(6) = 1/6 × 1/6 = 1/36
  • A. 1/36 — Correct: 1/6 × 1/6, because the two rolls are independent.
  • B. 1/12 — The fractions were added instead of multiplied. "And" multiplies, "or" adds.
  • C. 1/6 — That is the probability of one six, not of two in a row.
  • D. 1/18 — A halving slip. There are 36 equally likely outcomes, and only one of them is six and six.

Examiner technique: Ask whether the two events are joined by "and" or "or". And multiplies; or adds, after you check nothing overlaps.