Understanding the Probability of Shared Birthdays in a Given Contest

The question of how many people need to be in a room for there to be a high probability that two of them share a birthday has puzzled mathematicians for decades. In this context, we explore the probability that at least two out of 180 participants, competing during the dates of June 1 to June 23, will share the same birthdate. Let's break down this problem and understand why the probabilities vary.

The Misinterpretation of the Problem

First, it's important to note that the initial problem as it was conceptualized can be misinterpreted. The key point here is that the distribution of the 180 participants across the 23 days of the contest is not specified. If this distribution were even, the probability of shared birthdays would differ significantly from the case where such an even distribution is not assumed.

The Actual Probabilities and Calculations

Given that the exact distribution is not fixed, let's consider the probability that at least two participants share the same birthdate. To do this, we will use the complement method and the pigeonhole principle.

Using the Complement Method

The complement method involves calculating the probability that no two people share a birthday and then subtracting this from 1 to find the probability that at least two people share a birthday. The probability that no two people share a birthday can be calculated as follows:

No shared birthday: On the first day, any of the 365 days can be chosen. On the second day, any of the remaining 364 days can be chosen. On the third day, any of the remaining 363 days can be chosen.

Therefore, the probability that no one shares a birthday is:

P(0 shared birthdays) (365/365) * (364/365) * (363/365) * ... * (365 - (n-1)/365)

If there are 23 days in total, this becomes:

P(0 shared birthdays) (365/365) * (364/365) * (363/365) * ... * (343/365)

The probability that at least one pair shares a birthday is then:

P(at least one shared birthday) 1 - P(0 shared birthdays)

The Pigeonhole Principle

The pigeonhole principle states that if you have more items than containers and you put each item into a container, at least one container must contain more than one item. In this context, if there are more participants (180) than days in the contest (23), then by the pigeonhole principle, at least one day must have two participants with the same birthdate.

Practical Implications and Examples

If we disregard the exact distribution and assume a more uniform spread, the probability of at least two people sharing a birthday becomes astronomically high. For example, if we use the birthday problem for a fixed number of participants (n 180) and days (23), the probability can be calculated as:

P(at least one shared birthday) 1 - (365/365) * (364/365) * ... * (365 - (180-1)/365)

This calculation simplifies to:

P(at least one shared birthday) 1 - {3/4364363...187186/365^179 1/4365...187/366^179}

Given this, the probability is approximately 98%, indicating that it is extremely likely that at least two participants will share a birthday.

Other Considerations

The problem of shared birthdays, while fascinating, isn't always as precise as it appears. If we consider Pete Buttigieg, an actual example of a person sharing a birthday with someone else, the probability of such an event occurring is 100%. In a world with over 7 billion people, the chances of finding at least two people sharing a birthday are virtually certain.

According to the pigeonhole principle, with more participants than available days, at least one day will surely have multiple birthdays. This principle is a fundamental concept in combinatorics and probability theory.

Conclusion

In conclusion, the probability that at least two out of 180 participants will share the same birthdate during the June 1 to June 23 contest is very high, essentially 100%. This is because the number of participants far exceeds the number of days available, making shared birthdays inevitable under the pigeonhole principle.