 Solution

The events $B\cap A_i$ are pairwise disjoint and their union is $B$. Finite additivity and the definition of conditional probability therefore give the law of total probability:For the birthday problem, let $D$ mean that all birthdays differ. For $n\le 365$, sequential conditioning on previous distinct birthdays and independence of random variables giveHere each factor uses the stated bound $1-x\le e^{-x}$. For $n\ge 29$, $n(n-1)\ge 812>730\log 3$, soFor $n>365$ a match is certain by the pigeonhole principle. The threshold from the exponential bound is $(1+\sqrt{1+2920\log 3})/2\approx 28.8$, so $29$ suffices.