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Conditional Probability

Probability

Where the branches already do it

  • On a tree diagram the second set of branches is already conditional, because those probabilities depend on which first branch was taken
  • A "without replacement" question is therefore a conditional probability question, whether or not it says so
    • From a bag of 5 red and 3 blue counters, if the first taken is red then 4 reds remain out of 7, so P(second red, given first red) = 4/7
  • Reading such a value straight off the tree needs no calculation at all
A without-replacement tree for two counters drawn from a bag of 3 blue and 8 red: the first branches 3 over 11 and 8 over 11, and the second set of branches uses denominators of 10 with the counts adjusted, so the second probabilities depend on what was drawn first
Source: Combined conditional probabilities by Save My Exams
  • Where the condition refers to the second stage instead, the paths have to be compared: find the probability of the paths that satisfy both, then divide by the probability of all the paths satisfying the condition
Worked example

Reading a condition off a tree

A bag holds 5 red and 7 blue counters. Two are taken without replacement. Given that the first counter is red, find the probability that the second is also red.

Solution:

  • The condition fixes what has already happened: the first counter was red
  • After it is removed, 11 counters remain and 4 of them are red
  • P(second red, given first red) = 4/11
  • That is exactly the number written on the second-stage branch of the tree diagram
  • No formula is needed: a tree diagram's second-stage probabilities are already conditional
  • Multiplying 5/12 by 4/11 would answer a different question — the probability that both are red, which is 5/33