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very good question

(a) If there are k people in a room, then

P(at least 2 persons are born in same month) = 1 - P(all peope are born in different months) = $1 - \frac{12*11*\ldots (12-(k-1))}{12^k}$

So for 5 persons, it is $P_1 = 1 - \frac{12*11*10*9*8}{12^5}$, for 10 persons it is  $P_2 = 1 - \frac{12*11*10*9*8*7*6*5*4*3}{12^10}$, and for 15 people, it has to be $P_3 = 1$, because there are only 12 months and 15 persons, so atleast 2 persons must have birthday in same month.

So P(at least 2 persons are born in same month) = $\frac{1}{4}*P_1 + \frac{1}{4}*P_2 + \frac{1}{2}*1 \approx 0.903$

(b) Suppose event $S$ denotes that atleast two people are born in same month. Then

$P(k=10 | S) = \frac{P(S | k = 10)*P(k=10)}{ P(S)} = \frac{P_2*0.25}{0.903} \approx 0.2757$

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