5.4-1

5.4-1 How many people must there be in a room before the probability that someone
has the same birthday as you do is at least 1/2? How many people must there be
before the probability that at least two people have a birthday on July 4 is greater than 1/2?

The probability that someone has same birthday as me = 1 -(probability every person in the room has a different birthday as me) (every person can have birthday on other 364 days and it doesn't matter if they have birthday on same hence the 364 ) $$= 1 - (364/365)(364/365)...(364/365)$$ $$ = 1 - (364/365)^n = 1/2$$ $$ 1 - (364/365)^n \geq 1/2 $$ $$ (364/365)^n \leq 1/2 $$ applying logarithm on both sides $$ n lg(364/365) \leq lg(1/2) $$ $$ n lg(364/365) \leq - lg(2) $$ $$ n lg(365/364) \geq lg(2) $$ $$ n \geq lg(364/365) lg(2) $$ $$ n \geq 253 $$ (using a calculator ) For the second part the probability of at least two people have a birthday on July 4 is greater than 1/2 = 1 - probability of exactly one person born on july4 - probability that no one in the room is born on july 4 let probability of exactly one person born on july4 we can use binomial theorem here $$ {n \choose k} p^k (1-p)^{ n-k} $$ Let k be the no of people in the room(n=k) since we want exactly one person born on july 4(k =1). The probability of person born on a day in a year is 1/365(p =1/365) $$ {k \choose 1} (1/365)^k (1-1/365)^{ k-1} $$ $$ {k \choose 1} (1/365)^k (364/365)^{ k-1} $$ $$ {k \choose 1} (1/365)^k (364/365)^{ k-1} $$ probability that no one in the room is born on july 4: $$ (364/365)^k $$(from first part of analysis) probability of at least two people have a birthday on July 4 = $$1 - {k \choose 1} (1/365)^k (364/365)^{ k-1} - (364/365)^k $$ $$1 - {k \choose 1} (1/365)^k (364/365)^{ k-1} - (364/365)^k /geq 1/2 $$ If $k geq 613 $ the above equation is satisfied.