# Alteration of Birthday Problem

• mXSCNT
In summary, the conversation discusses the problem of finding the expected number of distinct birthdays in a room with k people on a planet where each year has n days. The formula provided is j = n - n * (1-1/n)^k, taking into account the probability of a given day being nobody's birthday. Another formula is suggested, taking into account the total number of possible birthdays and the probability of them being represented in the room. The conversation also explores the use of X and Xi to represent the number of distinct birthdays and the probability of at least one person having their birthday on a given day. The final answer is determined to be nk/(n+k-1).
mXSCNT
This is a restatement of the vocabulary problem which I introduced in https://www.physicsforums.com/showthread.php?t=293553. Perhaps these terms will be more familiar/less ambiguous.

Suppose we are on a planet where each year has n days, and in a room with k people. If birthdays are uniformly distributed throughout the year, how many distinct birthdays, j, do we expect to find in the room? (Alternatively, how many birthdays n-j are NOT represented in the room, on average?)

Here's the answer (someone else's idea): j = n - n * (1-1/n)^k. The probability that a given day is nobody's birthday is (1-1/n)^k, so the expected number of days that are nobody's birthday is n * (1-1/n)^k.

I think that, since its distinct birthdays (from what I understand, distinct bdays are only the ones that don't coincide on the same day), you will have nCk bdays total, thus the expected number on any given day would be (nCk)/n since they're uniformly distributed.

There will also be nC(n-k) bdays that don't happen, once again the expected number would be nC(n-k)/n since they're uniform. I'm not sure about this, but I think its intuitive.

Let X be the number of distinct birthdays, and for $1 \leq i \leq n$ define Xi to be 1 if there's at least one person whose birthday is on the ith day, and 0 otherwise. Then X = X1 + X2 + ... + Xn, so:

$$E(X) = \sum _{i=1} ^n E(X_i) = nE(X_1)$$

E(X1)
= Prob(at least one person has their birthday on day 1)
= 1 - Prob(no one has their birthday on day 1)
= 1 - (# of ways to arrange k birthdays amongst n-1 days, allowing repetition)/(# of ways to arrange k birthdays amongst n days, allowing repetition)
= $1 - \binom{n+k-2}{k} / \binom{n+k-1}{k}$

So the final answer is:

$$\frac{nk}{n+k-1}$$

AKG said:
define Xi to be 1 if there's at least one person whose birthday is on the ith day, and 0 otherwise.
But doesn't the question as for distinct birthdays, ie Xi would be 1 if there is only one birthday on day i and 0 if there is not only one birthday on day i?

## What is the "Alteration of Birthday Problem"?

The "Alteration of Birthday Problem" is a mathematical problem that calculates the likelihood of two or more people sharing the same birthday in a group of any size. It is based on the assumption that birthdays are evenly distributed throughout the year.

## How is the "Alteration of Birthday Problem" different from the original "Birthday Problem"?

The original "Birthday Problem" only considers the probability of two people sharing the same birthday in a group, while the "Alteration of Birthday Problem" looks at the probability of three or more people sharing the same birthday in a group.

## What factors affect the probability in the "Alteration of Birthday Problem"?

The main factor that affects the probability in the "Alteration of Birthday Problem" is the size of the group. As the group size increases, the probability of three or more people sharing the same birthday also increases.

## How is the probability calculated in the "Alteration of Birthday Problem"?

The probability in the "Alteration of Birthday Problem" is calculated using the formula P = 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 of n people.

## What are some real-life applications of the "Alteration of Birthday Problem"?

The "Alteration of Birthday Problem" has various applications in fields such as statistics, probability, and cryptography. It is also used in event planning to avoid scheduling conflicts and in genetics to analyze the probability of shared birthdays among family members.

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