Problem with bayesian updating example

Hi everybody,

I have an example in bayesian statistics which I've worked on for some hours now, without significant success. The problem is the following (all of this is structured and prepared in the attached Excel sheet):

There are two outcomes (W and L) and three signals for these outcomes (B, H, S). The following is known:
P(B)=0.6; P(H)=0.3, P(S)=0.1, P(W|B)=0.7; P(W|H)=0.25, P(W|S)=0.5.

Now I would like to calculate the other probabilities (i.e. P(W), P(B|W), etc.) - which I think I have managed to do. Furthermore, I want to calculate the updated probabilities after I have observed one signal of B, which I have so far failed to succeed at.

I have attached an Excel sheet that contains the data and my results so far.

Thanks for your help!



TS Contributor
Since \( \{B, H, S\} \) forms a partition, we have

\( P(W) = P(W|B)P(B) + P(W|H)P(H) + P(W|S)P(S) \)

And from the Bayes Theorem:

\( P(B|W) = \frac {P(W|B)P(B)} {P(W)} \)

Thanks for your reply! I also have:

P(L) = 1 - P(W),

P(L|B) = 1 - P(W|B), P(L|H) = 1 - P(W|H), P(L|S) = 1 - P(W|S),

P(B|L) = P(L|B)P(B) / P(L).

But now how do I update all of these probabilitites after having observed one or two independent draws of B?