# Variation of Birthday problem - Group of n people

I know this has been posted several times and I have gone through most of the relevant posts. Here is one which I am having a difficult time to solve:

There are 450 people in a room; (1) how many of them are expected to have the same birthday with some other person in the room, (2) with at least 2 other people in the room and (3) with at least 3.

(1) is easy - by the pigeonhole principle, 450-365 (or 366) = 85 people are expected to have the same birthday.

How do we do (2) and (3)?

I am thinking that in 85 people we have $$\frac {85*84} {2} = 3570$$ possible pairs so the probability for a 3rd person to share one of their birthdays is $$1-\frac {364}{365}^{85}$$. And then how do we find the expected number of people for each case?

Any help is greatly appreciated! Thank you!

• I don't understand your use of the pigeonhole principle. Use Linearity of expectation instead, with an indicator variable for each person in the room.
– lulu
Nov 3, 2020 at 9:20
• From the pigeonhole principle you get that at-least $86$ people share their birthdays (not necessarily on the same day). You could also have $100$ people sharing a birthday with some probability. You need to find the expected number of people who share a birthday, not the least. Nov 3, 2020 at 9:26

Hints:

• Picking a particular individual, what is the probability that person shares their birthday:

• with no other people
• with exactly one person
• with exactly two people?
• So for that individual, what is the probability that person shares their birthday:

• with at least one person
• with at least two people
• with at least three people?
• And using the linearity of expectation, what is the expected number of people that share their birthday

• with at least one person (much more than or $$85$$ or $$86$$)
• with at least two people
• with at least three people?

Following the hints:

• Picking a particular individual, the probability that person shares their birthday:

• with no other people is $$\frac{364^{449}}{365^{449}}$$
• with exactly one person is $${449 \choose 1}\frac{364^{448}}{365^{449}}$$
• with exactly two people is $${449 \choose 2}\frac{364^{447}}{365^{449}}$$
• So for that individual, the probability that person shares their birthday:

• with at least one person is $$1-\frac{364^{449}}{365^{449}}$$
• with at least two people is $$1-\frac{364^{449}}{365^{449}}-{449 \choose 1}\frac{364^{448}}{365^{449}}$$
• with at least three people is $$1-\frac{364^{449}}{365^{449}}-{449 \choose 1}\frac{364^{448}}{365^{449}} - {449 \choose 2}\frac{364^{447}}{365^{449}}$$
• And using the linearity of expectation, the expected number of people that share their birthday

• with at least one person is $$450\left(1-\frac{364^{449}}{365^{449}} \right)$$
• with at least two people is $$450\left(1-\frac{364^{449}}{365^{449}}-{449 \choose 1}\frac{364^{448}}{365^{449}} \right)$$
• with at least three people is $$450\left(1-\frac{364^{449}}{365^{449}}-{449 \choose 1}\frac{364^{448}}{365^{449}} - {449 \choose 2}\frac{364^{447}}{365^{449}} \right)$$

and these values are about $$318.7$$ (much more than or $$85$$ or $$86$$) and $$156.8$$ and $$57.1$$

• Henry: For the 1st case (probability someone shares his birthday with no other people): Isn't it zero? Nov 3, 2020 at 10:15
• @Sal.Cognato It is much more than zero: the first three sub-bullets come from a binomial distribution with parameters $450-1=449$ and $\frac{1}{365}$, so the probability a particular person shares their birthday with no other people is $\left(\frac{364}{365}\right)^{449} \approx 0.29176$ Nov 3, 2020 at 10:25
• Thank you! So the expected number of people who share their birthday with 1 more, is $\frac {1}{1-0.29176}$ = 1.41? Nov 3, 2020 at 11:02
• Strange! I would also expect that in any group of people >365, the probability is 100% but I think the tricky point is that this must apply for any person, right? Nov 3, 2020 at 11:17
• @TomGalle No - the probability a individual shares their birthday with at least one other person is $1-0.29176$ and the expected number of people who share their with at least one other person is $450$ times this Nov 3, 2020 at 11:23