Probability of Having the Same Birthday

[tryingtoexcelatmath]

Question: What is the probability of 2 people having the same birthday in the room?
There are 50 people in the room, we are only looking at regular years, 365 days, not leap years.

Solution Provided: Using the subtraction principle: 1 - ((365 * 364 * ... * 316) / (365^50))

My Question: the numerator looks like it is using permutations nPr, but shouldn't the numerator be using combinations nCr?

If for example we only look at 3 people in the room, we could get Jan 1, 2020, Mar 1, 2020, and Dec 1, 2020.
But we only want people with different birthdays, so wouldn't that be the same as getting Dec 1, 2020 then Mar 1, 2020 then Jan 1, 2020?

To me it seems like order does not matter, thus shouldn't the top be nCr instead of nPr?

 

[Deleted member 4993]

Question: What is the probability of 2 people having the same birthday in the room?
There are 50 people in the room, we are only looking at regular years, 365 days, not leap years.

Solution Provided: Using the subtraction principle: 1 - ((365 * 364 * ... * 316) / (365^50))

My Question: the numerator looks like it is using permutations nPr, but shouldn't the numerator be using combinations nCr?

If for example we only look at 3 people in the room, we could get Jan 1, 2020, Mar 1, 2020, and Dec 1, 2020.
But we only want people with different birthdays, so wouldn't that be the same as getting Dec 1, 2020 then Mar 1, 2020 then Jan 1, 2020?

To me it seems like order does not matter, thus shouldn't the top be nCr instead of nPr?

The solution of the problem above does not require application of "permutation" and/or "combination".

Choose any person (1) from the room.

Choose another person (2) from the room. What is the probability that (1) and (2) does NOT have the same birthday - \(\displaystyle \frac{364}{365}\)

Choose another person (3) from the room. What is the probability that (1), (2) and (3) does NOT have the same birthday - \(\displaystyle \frac{364*363}{365^2}\)

Choose another person (4) from the room. What is the probability that (1), (2), (3) and (4) does NOT have the same birthday - \(\displaystyle \frac{364*363*362}{365^3}\)

If you have 23 or more people in the room - the probability of matching a birthday goes above 50%.