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3.14

# Bayes' theorem

Bayes’ theorem is one of the most useful results in probability, and (among other things), enables us to adjust our estimate of the probability of an event in the light of new evidence.

A full discussion of this topic is beyond our scope, although the relevant article on Wikipedia provides a useful summary – and you can of course use the comments to go into this topic more deeply. For our purposes, we mainly need to know how to deal with problems of the types exemplified below. (We have already encountered problems of this type – when we worked on ‘The Dog Ate my Homework’ in Week 2 of the course.)

## Sample examination-style questions

### Question 6

50 students are asked about their families: 30 have at least one brother, 25 have at least one sister, and 5 have neither a brother nor a sister.

• Represent this information on a Venn diagram.

• If I pick a student at random, what is the probability they have both a brother and a sister?

• If I pick a student at random and they have a brother, what is the probability they also have a sister?

### Question 7

Bag A contains one blue and two green counters. Bag B contains four blue counters and one green counter. I pick a bag at random, and take out a counter. If I draw a blue counter, what is the probability I have chosen Bag A?