Combined-Probability Aqa Foundation

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Exam code:8300

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)

What does independent mean?

Independent events are events that do not affect each other
- e.g. the probability of rolling a 6 on a fair dice and the probability of getting a head when flipping a coin
Be careful: questions ‘without replacement’ are not independent
- e.g. the probability of taking a red card out of a pack, not replacing it, then finding the probability of taking a second red card out of the same pack
  - The first event affected the number of cards left for the second event

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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