Question

There are 4 people in a room. What’s the probability that there are two people born on the same day of the week? (Assume all birthdays are independent and are uniformly distributed over the seven days of the week.)

Answer 1

There are 4 people in a room.

We have to find the probability that there are two people, born on the same day of the week.

Now, there are 7 days in the week.

So, the first person can have birthday on any 1 of the 7 days.

For any of these 7 choices, the second person can independently have birthday on any 1 of the 7 days.

Similarly, the third person can have birthday on any 1 of the 7 days.

And, similarly, the fourth person can have birthday on any 1 of the 7 days.

So, the 4 persons can have their birthdays, in 7*7*7*7, ie. 2401 number of possible ways.

So, all possible cases is 2401.

Now, we have to find the chance that exactly two people have their birthdays, on the same day of the week.

Now, first we choose which 2 people would have their birthdays on the same day of the week; this 2 people can be chosen out of 4, in

  

or number of ways.

Now, for any of these 6 choices, these two people can have their birthday on the same day, in any 1 of the 7 days of the week.

The third people can have birthday on 6 days of the week; and the last one can have birthay on 5 days of the week.

So, the number of favourable cases is

6*7*6*5

or, 1260.

So, the probability that there are exactly two people having the same birthday, is

So, the required probability is 0.525.