GCSEMathsWJECMaths & Numeracy (Double Award)HigherRevision NotesStatistics & ProbabilityProbability of Multiple EventsMutually Exclusive & Independence

Mutually Exclusive & Independence (WJEC GCSE Maths & Numeracy (Double Award)): Revision Note

Exam code: 3320

Jamie Wood

Written by: Jamie Wood

Reviewed by: Mark Curtis

Updated on

Mutually Exclusive & Independence

What are mutually exclusive events?

  • Two events are mutually exclusive if they cannot both happen at the same time

    • In one go / attempt / trial of an experiment, if the first event happens, then the second event cannot happen

      • e.g. when rolling a dice, the events “getting a prime number” and “getting a 6” are mutually exclusive (as 6 is not prime)

  • If A and B are mutually exclusive events, then

    • the probability of A and B happening is zero

    • the probability of either A or B happening is found by adding

      • P(A) + P(B)

  • The complement of event A is the event where A does not happen

    • Complementary events are mutually exclusive

    • Probabilities are related by

      • P(complement of A) = 1 - P(A)

    • e.g. if P(yellow) = 0.3 then P(not yellow) = 1 - 0.3 = 0.7

Worked Example

A bag contains 20 counters: 5 Red (R), 8 Blue (B), and 7 Yellow (Y).

A single counter is drawn from the bag.

Calculate the probability of selecting either a Red counter or a Blue counter on a single draw.

Answer:

The events "selecting a red counter" and "selecting a blue counter" on the same draw are mutually exclusive (the counter cannot be both colours)

Add their probabilities together

5 over 20 plus 8 over 20

P(red or blue) = 13 over 20

What are independent events?

  • Two events are independent if one occurring does not affect the probability of the other occurring

    • For example: when flipping a fair coin twice the events “getting a tails on the first flip” and “getting a tails on the second flip” are independent

  • If A and B are independent events then

    • The probability of A and B occurring is P(A) x P(B)

  • This result can be used to test whether two events are independent or not

    • i.e. A and B are only independent if the probability of them both happening is exactly equal to P(A) x P(B)

Worked Example

A standard unbiased 6-sided dice is rolled twice.

Calculate the probability of rolling a 4 on the first roll and an even number on the second roll.

Answer:

Rolling a 4 on the first roll does not affect the probability of rolling an even number on the next roll

These events are independent

Therefore we can use the result "the probability of A and B occurring is P(A) x P(B)"

The probability of rolling a 4 on the first roll is 1 over 6

The probability of rolling an even number (2, 4, 6) on the second roll is 3 over 6 equals 1 half

The probability of "rolling a 4 on the first role" and "rolling an even number on the second roll" is equal to P(A) x P(B)

1 over 6 cross times 1 half

P(rolling a 4 then rolling an even) = 1 over 12

Examiner Tips and Tricks

In the previous example, 'rolling a 4 on the first roll" and "rolling an even number on the second roll" cannot be mutually exclusive as they happen on different trials of the experiment.

Worked Example

A teacher records data for 20 students about two events:

Event A: The student revised the night before a test.

Event B: The student scored at least 70%.

The results are summarised in the two-way table below.

Scored at least 70% (B)

Scored less than 70% (not B)

Total

Revised (A)

7

3

10

Did not revise (Not A)

4

6

10

Total

11

9

20

Determine if the events A and B are independent or not.

Answer:

If A and B are independent events then the probability of A and B occurring is P(A) x P(B)

Find P(A) which is the probability that the student revised the night before the test

straight P open parentheses straight A close parentheses equals 10 over 20

Find P(B) which is the probability that the student scored at least 70%

straight P open parentheses straight B close parentheses equals 11 over 20

Find the probability of A and B occurring by looking at the right place on the table

These are the students who revised and scored at least 70%

straight P open parentheses straight A space and space straight B close parentheses equals 7 over 20

Now test if 7 over 20 is equal to the calculation P(A) x P(B)

straight P open parentheses straight A close parentheses cross times straight P open parentheses straight B close parentheses equals 10 over 20 cross times 11 over 20 equals 11 over 40

This is not equal to 7 over 20

Make a conclusion

Events A and B are not independent as 7 over 20 does not equal 10 over 20 cross times 11 over 20

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.