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Combined Probabilities (Cambridge (CIE) IGCSE International Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Cambridge (CIE) International Maths: CoreCambridge (CIE) Maths: ExtendedCambridge (CIE) Maths: CoreEdexcel Maths A: Higher

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Combined Probability

How do I calculate combined probabilities?

  • You can calculate probabilities of one event after another without needing tree diagrams

    • These are called combined (or successive) probabilities

  • There are two rules to learn

    • And means multiply and or means add

    • P(A and B) = P(A) x P(B)

    • P(AA or BB) = P(AA) + P(BB)

  • Try to rephrase each question using and / or

    • For example, when flipping a coin twice: 

      • P(two heads) = P(head and head)

      • P(both the same) = P(head and head or tail and tail) = P(HH) + P(TT)

  • Remember that P(not A) = 1 - P(A)

Worked Example

A box contains 3 blue counters and 8 red counters.
A counter is taken at random and its colour is noted.
The counter is put back into the box.
A second counter is then taken at random, and its colour is noted.

Work out the probability that

(a) both counters are red,

P(both red) = P(red and red) 
This is P(red) × P(red) using the 'and rule'

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 8 over 11 cross times 8 over 11 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold 64 over bold 121 end cell end table

(b) the two counters are of different colours.

P(different colours) = P(blue and red or red and blue)
This is P(B and R) + P(R and B) using the 'or rule'
This is P(B) × P(R) + P(R) × P(B) using the 'and rule' twice

table row blank blank cell 8 over 11 cross times 3 over 11 plus 3 over 11 cross times 8 over 11 end cell row blank equals cell 24 over 121 plus 24 over 121 end cell end table

table row blank blank cell bold 48 over bold 121 end cell end table

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    Previous:Probability Tree DiagramsNext:Conditional Probability
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.