GRAPHS AND STATISTICS

Combined Probability

Learn the reliable method for combined probability, then apply it to 11+ exam-style questions.

1 · WHAT YOU NEED TO KNOW

The idea

Combined probability tracks the possible outcomes across more than one event.

Picture itCount favourable and possible outcomes
2 · THE METHOD

Follow these steps

  1. List the outcomes or draw a simple tree.
  2. Multiply along independent branches and add mutually exclusive successful outcomes.
  3. Check whether replacement changes the second probability.
3 · WORKED EXAMPLES

See the method in action

Example 1

A bag holds 4 red, 2 blue and 3 green counters. One counter is chosen at random. What is the probability that it is red or blue?

  1. There are 6 red or blue counters out of 9 counters altogether, so the probability is 2/3.
Answer: 2/3
Example 2

A bag contains 4 red and 5 blue counters. A counter is chosen, replaced, and another is chosen. What is the probability that both counters are red?

  1. With replacement, each red probability is 4/9.
  2. Multiply: 4/9 × 4/9 = 16/81.
Answer: 16/81
4 · ELEVENSPRINT TIP

A useful shortcut

Check whether replacement changes the second probability.

5 · COMMON MISTAKES

Watch out for these

6 · TRY IT YOURSELF

Three fresh questions

Ready to check the method? You will get three fresh, validated questions.