In a Leap year a month is selected at random and a day is selected at random and found that its fifth Friday. What is the Probability that selected month has $30$ days.
My try: Let $A$ be an event of day chosen is fifth friday
$M_{30}$ be an event of chosen month having $30$ days
$M_{29}$ be an event of chosen month having $29$ days
$M_{31}$ be an event of chosen month having $31$ days
$P(M_{30})=\frac{4}{12}$
$P(M_{29})=\frac{1}{12}$
$P(M_{31})=\frac{7}{12}$
we need to find
$$P\left(M_{30}/A\right)$$
By Bayes theorem we have
$$P\left(M_{30}/A\right)=\frac{P\left(A/M_{30}\right)P(M_{30})}{\sum P(A)}$$
but how to find $P\left(A/M_{30}\right)$?
I think you are on the right track.
Now, I'd define $B$ as the event : "the month has a fifth friday".
Then $$ \begin{align}P(A \mid M_i) &= P( A, B \mid M_i) + P( A, B^c \mid M_i) \\ &= P( A, B \mid M_i) \\ &= P( A \mid B ,M_i) \, P(B \mid M_i) \end{align}$$
Can you go on from here?