Three friends find out their birthdays are all within the same week.
Supposing their birthdays are otherwise random, what is the probability they all have their birthday on the same day?
Logical Approach : Since its given that birthdays are in same week the sample space is reduced to 7 possibilities for a person. Hence total possibilities = 7^3 Favorable cases = 7*1*1. So answer = 1/49
If I go by formula, let A = probability three have birthday in same day B = probability three have birthday on same week Then we have to find P(A|B) = P(A∩B)/P(B) My question is how do I calculate this if I go by this method?
First, note that $P(A\cap B)=P(A)=(365*1*1)/365^3=1/365^2$ and $P(B)=(52*1*1)/52^3=1/52^2$. Therefore, you get $$P(A|B)=\frac{(1/365^2)}{(1/52^2)}=\frac{52^2}{365^2}=\left(\frac{52}{365}\right)^2\approx\left(\frac{1}{7}\right)^2=\frac{1}{49}$$