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Probability of combined events Cambridge IGCSE Mathematics (9–1) revision

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In plain words

What's the chance of two things both happening? A tree diagram lays out every route. The rule is simple: multiply along the branches, and add the routes that give what you want.

Three things to know

  1. For "and" (both happen), multiply the probabilities. For "or" (one or the other of things that can't both happen), add them.
  2. On a tree diagram, multiply along a route and add the answers for the different routes. The branches from any one point add up to 1.
  3. If something is taken and not put back, the probabilities change for the second pick: there is one fewer altogether, and one fewer of whatever was taken.

Worked example

A bag has 3 red and 2 blue balls. Two are taken without replacement. Find the probability that both are red.

  1. First red: 3 out of 5.
  2. One red has gone, so second red: 2 out of 4.
  3. 3/5 × 2/4 = 6/20 = 3/10.

Tips and tricks

  • "Without replacement": both the top and the bottom of the second fraction go down.
  • For "at least one", it's quicker to work out 1 − P(none).
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Probability of combined events: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. Two events are independent with probabilities 1/2 and 1/3. What is the probability that both happen?
    • 5/6
    • 1/6 (the answer)
    • 2/5
    • 1/5

    Multiply for "and".

  2. On a tree diagram, what do you do along a route?
    • add
    • multiply (the answer)
    • subtract
    • divide

    You add only between different routes.

  3. A dice is rolled. What is the probability of a 1 or a 2?
    • 1/36
    • 1/6
    • 1/3 (the answer)
    • 1/2

    1/6 + 1/6, since both can't happen at once.

  4. A bag has 2 red and 3 blue balls. Two are taken without replacement. What is the probability that both are red?
    • 4/25
    • 1/10 (the answer)
    • 2/5
    • 1/5

    2/5 × 1/4.

  5. The probability of rain on each of two days is 0.5, independently. What is the probability of rain on at least one day?
    • 0.25
    • 0.5
    • 0.75 (the answer)
    • 1

    1 − P(no rain on either day) = 1 − 0.25.

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