00:01
And this problem is said that if n people are present in a room, then what is the probability that at least two of them celebrate their birthday on the same day of the year? now, first of all, let us find the probability that none of the people, none of these n people, celebrate their birthday on the same day of the year.
00:28
So none of the people have the same birthday.
00:33
Let's find this probability first.
00:38
So this will be the number of favorable outcomes divided by the total number of outcomes.
00:46
So first of all, let us consider the total number of outcomes, which we need to write in the denominator.
00:51
So we have n people, and we are considering the birthdays of n people.
00:58
So one person will have one birthday out of the 365 days.
01:05
In a year.
01:06
So there are 365 possible options for the birthday of the first person.
01:12
Similarly, there will be 365 options for the second person's birthday, 365 options for the third person's birthday, and so on until we get to n.
01:21
So the total number of options for choosing the birthdays of n people will be 365 to the power n.
01:30
This is because we have 365 for the first person, 365 for the second person and so on.
01:36
So we use the multiplication rule of counting and we get 365 times 365 times 365 so that happens n times so that we end up with 365 to the power end be the total number of ways in which we can choose birthdays for these end people.
01:52
So that's the total number of outcomes.
01:54
On the numerator we need to write the number of favorable outcomes.
01:59
I'm considering that none of the people have the same birthday.
02:02
So what we need to need to do is out of the 365 days of the year, we need to choose n dates for the birthdays.
02:12
We need to have exactly n different dates selected for the n people.
02:17
And that way they will all have different birthdays.
02:20
And the number of ways we can do it in this case would be 365 p .n.
02:24
Here we use p, which represents permutation, because in this case, the order of selection matters.
02:31
This means that suppose if this is the post person, this is the second person, and we go up to this nth person.
02:37
And maybe we assign january 1 to this person.
02:42
Maybe we give february 2 to this person, and maybe this person gets september 8...