Birthday Problem (At most 3 share a birthday)

• TaliskerBA
In summary, the challenge presented by the lecturer is to find the probability of at most 3 people sharing a birthday out of a group of N people, after already explaining how to calculate the probability of at least 2 people sharing a birthday. This can be solved recursively or as a finite-state Markov chain.
TaliskerBA
I have started a course on probability and the lecturer, after explaining how to work out the classic problem of the likelihood that at least 2 people share a birthday out of a field of N people (which I understood fine), issued the challenge of working out what the probability of at most 3 people sharing a birthday (out of N people) is. I have had a think about this and am at a loss at how to work this out. Can anyone offer any advice about how to tackle this problem?

Thanks,

Talisker

TaliskerBA said:
I have started a course on probability and the lecturer, after explaining how to work out the classic problem of the likelihood that at least 2 people share a birthday out of a field of N people (which I understood fine), issued the challenge of working out what the probability of at most 3 people sharing a birthday (out of N people) is. I have had a think about this and am at a loss at how to work this out. Can anyone offer any advice about how to tackle this problem?

Thanks,

Talisker

The probability that at most two people share a birthday would be:

P(no people share birthday) + P(exactly 1 pair of people share a birthday) + P(exactly 2 pairs of people share different birthdays) + ...

To get at most 3 people share a birthday just extend this to get:

$$\text{P(no people share birthday)} + \sum_{i=1}^{N/2} \text{P(exactly i pairs of people share different birthdays)} + \sum_{i=1}^{N/3} \text{P(exactly i triples of people share different birthdays)}$$

TaliskerBA said:
I have started a course on probability and the lecturer, after explaining how to work out the classic problem of the likelihood that at least 2 people share a birthday out of a field of N people (which I understood fine), issued the challenge of working out what the probability of at most 3 people sharing a birthday (out of N people) is. I have had a think about this and am at a loss at how to work this out. Can anyone offer any advice about how to tackle this problem?

Thanks,

Talisker

It could be solved recursively in terms of N. In solving the original problem the lecturer would have (implicitly at least) kept track of the number of days with 0 or 1 birthdays as one more person is added to the group. Same idea applies for the new problem, but including the number of days with 2 or 3 birthdays as well. There's more combinations to consider here but really it's a finite-state Markov chain so in principle the solution could be written as a matrix product.

What is the "Birthday Problem (At most 3 share a birthday)"?

The "Birthday Problem (At most 3 share a birthday)" is a mathematical problem that asks for the probability that, in a group of n people, at least 3 of them share the same birthday. It is also known as the "Birthday Paradox".

What is the significance of the "Birthday Problem (At most 3 share a birthday)"?

The "Birthday Problem (At most 3 share a birthday)" has real-life applications in fields such as probability, statistics, and cryptography. It is often used to demonstrate the counterintuitive nature of probability and the concept of large numbers.

How is the probability of the "Birthday Problem (At most 3 share a birthday)" calculated?

The probability of the "Birthday Problem (At most 3 share a birthday)" can be calculated using the formula 1 - (365!/(365^n*(365-n)!)), where n is the number of people in the group. This formula takes into account the number of possible combinations of birthdays in a group and subtracts it from 1 to give the probability of at least 3 people sharing a birthday.

What is the relationship between the number of people and the probability in the "Birthday Problem (At most 3 share a birthday)"?

The probability in the "Birthday Problem (At most 3 share a birthday)" increases as the number of people in the group increases. This is because the more people there are, the higher the chance of two or more of them sharing the same birthday. The probability reaches 100% when there are 366 people in the group, as there are only 365 possible birthdays.

Are there any variations of the "Birthday Problem"?

Yes, there are variations of the "Birthday Problem" that involve different numbers of people sharing a birthday, or different date ranges for the birthdays. These variations can be solved using similar probability calculations, but the results may vary.

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