# Probability that Month selected has $30$ days

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 $$P\left(A/M_{30}\right)$$?

• Assuming you are working with Gregorian calendar (which is a pretty reasonable assumption, unless you want to work with Julian calendar for e.g. astronomical reasons), use its minimal period of 400 years to calculate $\mathbf{P}(A\cap M_{30})$. – user10354138 Oct 14 '18 at 15:24

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}