1 votes 1 votes In a room of $23$ people what is the probability that $2$ of. them has a birthday in common? Probability probability birthday + – srestha asked Oct 4, 2018 srestha 793 views answer comment Share Follow See all 12 Comments See all 12 12 Comments reply Show 9 previous comments MiNiPanda commented Oct 4, 2018 reply Follow Share It is like what Somashree did.. prob(exactly two are having same birthday) : Say person 1 takes birth first and can choose to have birthday on any one of the 365 days. So 365 ways. Then person 2 comes and he has to choose that one day which person 1 has chosen. So he can only choose 1 day out of 365 and has no other choice. person 3 cannot choose the day chosen by them. So he can chose 364 days from 365 days leaving those 1 day for person 1 and person2. (364= 365-3+2) person 4 cannot choose the day chosen by previous 3. So he can chose 363 days from 365 days leaving those 2 days for person1,person2,person3. (363= 365-4+2) Likewise person 23 can take birth on (365-23+2) = 344 days. So no. of ways in which person 1 and person 2 only share same birthday is : 365x1x 364x363.....x344 Now how many pairs of people can be formed out of 23 people who can have same birth date? 23C2. I showed the case where only person1 and person2 share birthday and none of the other does share his birthday with any body else. But there can be 23C2 ways in which person i and person j can have same birthday and no other does. So final no. of ways = 23C2 x 365x1x 364x363.....x344 If no such restrictions were there then each person can take birth on any of the 365 days. So 365 ways for each of the 23 people. Reqd probability = 23C2x (365x1x364x363x.....344)/ (365x365.....x36523 times) = 0.36 (approx) 2 votes 2 votes Subarna Das commented Oct 5, 2018 reply Follow Share https://gateoverflow.in/220470/mission-iit-screening-test - Check this 0 votes 0 votes srestha commented Oct 5, 2018 reply Follow Share no , it is exactly 2 :) 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes therefore the probability is = 23C2 *(1/365)*(364*363*362*...*344)/(365)21 Or therefore the probability is = 23C2 *(1/365)^2*(364*363*362*...*344)/(365)21 Which one is correct@ srestha Raghav Khajuria answered Oct 5, 2018 Raghav Khajuria comment Share Follow See all 0 reply Please log in or register to add a comment.