Originally Posted by

**probabilityaddict** [snip]

4. (a) State and prove Bayes theorem for calculating posterior probabilities from prior probabilities and some observed events.

(b) A committee of 3 people has been formed by random selection from five left wingers and four right wingers. The committee members then vote for or against a strike whenever there is a dispute. Each left winger votes for a strike three our of four times in strike votes, whereas each right winger votes for a strike only once out of three times in strike votes. If it is known that the committee has decided against one particular strike, what is the probability that there is a majority of right wingers on the committee?

Does anyone know how to approach these?

(b) Let X be the random variable *number of right wingers on committee*.

Pr(X > 1 | committee decides against strike) = Pr( X > 1 and committee decides against strike)/Pr(committee decides against strike).

Pr(committee decides against strike) = Pr(committee decides against strike | X = 0) Pr(X = 0) + Pr(committee decides against strike | X = 1) Pr(X = 1) + Pr(committee decides against strike | X = 2) Pr(X = 2) + Pr(committee decides against strike | X = 3) Pr(X = 3).

I'll calculate Pr(committee decides against strike | X = 2) Pr(X = 2):

$\displaystyle \Pr(X = 2) = \frac{{4 \choose 2} {5 \choose 1} }{{9 \choose 3}} = \frac{5}{14}$.

Committe decides against a strike => either 2 or 3 votes against strike:

Pr(3 votes against) = $\displaystyle \left( \frac{1}{4}\right)^2 \left( \frac{2}{3}\right) = \frac{1}{24}$.

Pr(2 votes against) = $\displaystyle \left( \frac{1}{4}\right)^2 \left( \frac{1}{3}\right) + 2 \left( \frac{1}{4}\right) \left( \frac{3}{4}\right) \left( \frac{2}{3}\right) = \frac{13}{48}$.

Therefore Pr(committee decides against strike | X = 2) = $\displaystyle \frac{15}{48}$.

Pr( X > 1 and committee decides against strike) = Pr( X = 2 and committee decides against strike) + Pr( X = 3 and committee decides against strike)

= Pr(committee decides against strike | X = 2) Pr(X = 2) + Pr(committee decides against strike | X = 3) Pr(X = 3).

These will already have been calculated - above.

Note: I reserve the right for careless mistakes in this reply.