Probability of sharing a birthday

[locke]

I'm trying to figure out what the probability of two people in a room sharing a birthday is, if there are 19 people in the room.

Originally i assumed that the chances of any pair of people sharing a birthday are 1/365. Since there are 19c2 pairs of people in the room in question, I thought the probability was 19c2/365. This is obviously wrong though, since if the number of pairs in the room was >365, this type of reasoning would yield probabilities greater than one.

Can anyone point me towards the right answer?

 

[ceptimus]

Since you are concerned with whether two or more people share the same birthday, it's easier to calculate the probablilty that a birthday is not shared, and then subtract that from 1.

The probability that 4 people will all have different birthdays (ignoring leap years) is:

364/365 * 363/365 * 362/365

Explanation: We don't worry about the first person, as a single person can't have a shared birthday. The second person has 364 days to choose from, the third person 363, and the fourth person 362.

Extending this to 19 people, we get the probability that they all have different birthdays is:

(364! / (364 - (19 - 1))!) / 365^(19 - 1)

So the probability that two (or more) people share a birthday is one minus that.

1 - (364! / 346!) / 365^18 = 0.379119 (approximately)

 

[Architeuthis Dux]

The probability of two people in a room sharing a birthday is not as simple as calculating the number of pairs in the room and dividing by 365. This is because the probability of two people sharing a birthday depends on the number of people in the room and the number of possible birthdays they could share.

In this case, with 19 people in the room, the probability of two people sharing a birthday is actually much higher than 1/365. This is because there are 19 possible pairs of people who could share a birthday, not just one. So the probability would be 19/365, which is approximately 0.052 or 5.2%.

To calculate the probability more accurately, we can use the formula for the birthday problem, which is 1 - (365/365 * 364/365 * 363/365 * ... * (365-n+1)/365), where n is the number of people in the room. For 19 people, this would give us a probability of approximately 0.415 or 41.5%.

It's important to note that this is not a definitive answer, as the probability can vary based on factors such as leap years and the distribution of birthdays in the group. But it gives us a good estimate of the likelihood of two people sharing a birthday in a room of 19 people.