Sharing a birthday - what's wrong in my approach?

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Homework Help Overview

The discussion revolves around the classic birthday problem, specifically the question of how many people are needed in a group to achieve a 50% probability that at least two individuals share the same birthday. The original poster expresses a desire to develop their own method for solving this problem, despite acknowledging existing solutions.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to derive the solution by calculating the number of unique pairs in a group and applying a probability of 1/365 for shared birthdays. They express confusion over their result, which suggests 20 people yield a 52% probability, while established sources indicate 23 people are required for the same probability.
  • Some participants question the validity of adding probabilities directly and suggest that the original poster's approach may not account for the complexities of the problem, such as the need to consider the probability of no shared birthdays.
  • Others highlight that the problem involves more than just pairs and may require a different method to accurately assess the probability of shared birthdays among larger groups.

Discussion Status

The discussion is ongoing, with participants providing insights and questioning the assumptions made by the original poster. There is a recognition of the need for a more nuanced understanding of the problem, particularly regarding the calculation of probabilities in larger groups.

Contextual Notes

Participants note that the original poster's method may be an approximation that works for smaller groups but becomes less accurate as the number of individuals increases. The complexity of calculating probabilities for shared birthdays is acknowledged, indicating that the problem may require a more sophisticated approach than initially considered.

musicgold
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This is not really a homework problem. This is something I enjoy thinking about.

1. Homework Statement
How many people should there be in a group to have a 50% chance of at least two of them having the same birthday.
I know that there a many pages explaining how to solve this and the answer itself. But I wanted to develop my own way to figure this out.

Homework Equations


The problem boils down to how many unique pairs or handshakes can be formed in the group. That is N x (N-1) / 2 .
Also, the probability that a pair has the same birthday is 1/365.

Therefore, I think I have solve the following equation for N
N x (N-1) = 365

The Attempt at a Solution


The approximate value of N is 20. For N=20, I think the probability of having at least one pair with the same birthday is 52%. But according this Wikipedia page there has to be 23 people for a 50% probability in this case.

Clearly I am doing something wrong. Maybe assigning the 1/365 probability to every pair is not correct. I don't know.
 
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You can't add probabilities like that. Try to use the same approach to find when there is certainly a shared birthday and the problem should become obvious.
Similarly: 3 die rolls don't give you a 50% probability to roll a 1. The probability is a bit lower.

You calculated the number of people where we have on average 1/2 shared birthday.
 
musicgold said:
This is not really a homework problem. This is something I enjoy thinking about.

1. Homework Statement
How many people should there be in a group to have a 50% chance of at least two of them having the same birthday.
I know that there a many pages explaining how to solve this and the answer itself. But I wanted to develop my own way to figure this out.

Homework Equations


The problem boils down to how many unique pairs or handshakes can be formed in the group. That is N x (N-1) / 2 .
Also, the probability that a pair has the same birthday is 1/365.

Therefore, I think I have solve the following equation for N
N x (N-1) = 365

The Attempt at a Solution


The approximate value of N is 20. For N=20, I think the probability of having at least one pair with the same birthday is 52%. But according this Wikipedia page there has to be 23 people for a 50% probability in this case.

Clearly I am doing something wrong. Maybe assigning the 1/365 probability to every pair is not correct. I don't know.

The usual birthday problem looks at more than "pairs"; it asks for the probability that at least two people share a common birthday; perhaps three, or four, or more people share that same birthday. It does so by looking at the probability of no common birthdays, then taking its complement.

The problem of computing the probability that exactly one pair share a birthday (and no others do) is much harder.
 
Last edited:
musicgold said:
This is not really a homework problem. This is something I enjoy thinking about.

1. Homework Statement
How many people should there be in a group to have a 50% chance of at least two of them having the same birthday.
I know that there a many pages explaining how to solve this and the answer itself. But I wanted to develop my own way to figure this out.

Homework Equations


The problem boils down to how many unique pairs or handshakes can be formed in the group. That is N x (N-1) / 2 .
Also, the probability that a pair has the same birthday is 1/365.

Therefore, I think I have solve the following equation for N
N x (N-1) = 365

The Attempt at a Solution


The approximate value of N is 20. For N=20, I think the probability of having at least one pair with the same birthday is 52%. But according this Wikipedia page there has to be 23 people for a 50% probability in this case.

Clearly I am doing something wrong. Maybe assigning the 1/365 probability to every pair is not correct. I don't know.

Your method is an approximation for low probabilities. Let's take an example of 3 people, A, B, C. What is the probability of a shared birthday?

Your method is:

Probability A & B share a birthday = 1/365
Probability A & C share a birthday = 1/365
Probability B & C share a birthday = 1/365

So, total probability is 3/365 = 0.008219

But, that's only an approximation to the correct answer, which is

Probability A & B do not share a birthday = 364/365
Probability that C does not share a birthday with A & B (given that A & B do not share a birthday) = 363/365

Probability that A, B, C all have different birthdays = (364/365)(363/365).

Probability that there is a shared birthday = 1 - (364/365)(363/365) = 0.008204

As you increase the number of people, the approximation becomes less accurate (as you've discovered for about 20 people).
 

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