# Bayesian statistics example

I am learning Bayesian statistics for a data mining module. We were given a set of practice questions but no answers and I want to make sure that I am going about Bayes in the right way.

Suppose Manchester United were about to play Arsenal in the Premiership and you assess the probability of Manchester United winning to be 0.7. You also feel that if they did win, there is a probability of 0.9 that your local pub will be packed with fans celebrating their club’s victory. Alternatively, if they loose, you believe that there is still a probability of 0.6 that the pub will be packed with fans albeit for drowning their sorrows. As someone not interested in football, you enjoy a quiet day at work and on the way home, notice a big crowd in the pub. Can you assume that Manchster United won the game? If so, what is the probability of them having won the game?

Attempted solution:

let $p(w)$ denote prob of manchester united winning let $p(f)$ denote prob of pub being full

$p(w|f)=\frac{p(f|w)*p(w)}{p(f)}$ , where $p(f)=p(f|w)*p(w)+p(f|¬w)*p(¬w)$

$p(f)= 0.9*0.7+0.6*0.3=0.81$

$\rightarrow$$p(w|f)=\frac{0.9*0.7}{0.81}=0.78$

So prob of Manchester untied having won is 0.78.

I don't think we can assume they won just because there were people in the pub as there was a 0.6 prob of them drowning their sorrows.

Have I got this right or have I misunderstood Bayesian statistics?

Thanks very much

• Your answer is correct. – Cyan Apr 29 '12 at 21:53