The birthday problem concept question

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In summary, A(i) represents the event where the first i people have different birthdays, and A(i+1) is a subset of A(i) where the first i+1 people have different birthdays. The reason A(i+1) is a subset of A(i) is because A(i) also includes i+1 people who have the same birthday as one of the first i, making it a larger set.
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maiad
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Let A(i) be the event that the first ith person have different birthdays for i=1,2,3...,n.
We note that A(i+1) is a subset of Ai such that Ai(A(i+1))= A(i+1)

I wonder why A(i+1) is a subset of Ai. If the first 3 people have no birthdays in common, shouldn't that also mean the first 2 people doesn't either? By that logic, shouldn't Ai be the subset of A(i+1)?
 
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Your reasoning is correct up to the conclusion. Ai includes i+1 people who have same birthday as one of the first i, as well as all of A(i+1), therefore Ai is the bigger set.
 

FAQ: The birthday problem concept question

What is the birthday problem concept?

The birthday problem concept, also known as the birthday paradox, is a mathematical problem that explores the probability of two or more people sharing the same birthday in a group. It may seem counterintuitive, but in a group of just 23 people, there is a 50% chance that two people have the same birthday.

How is the probability of shared birthdays calculated?

The probability of shared birthdays is calculated using the formula P(n) = 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 get the probability of at least two people sharing a birthday.

Why is the probability of shared birthdays higher than expected?

The probability of shared birthdays is higher than expected due to the nature of the problem. Most people assume that the probability of two people sharing a birthday is low, but the calculation takes into account any possible combination of birthdays in a group rather than just comparing two specific individuals.

How is the birthday problem concept applicable in real life?

The birthday problem concept has many real-life applications, such as in cryptography, data analysis, and even in birthday planning. In cryptography, it can help assess the likelihood of two people having the same password or encryption key. In data analysis, it can be used to identify patterns or anomalies in large datasets. And in birthday planning, it can help predict the likelihood of shared birthdays in a group of guests.

Can the birthday problem concept be extended to more than two people sharing a birthday?

Yes, the birthday problem concept can be extended to more than two people sharing a birthday. The general formula for calculating the probability of k people sharing a birthday in a group of n people is P(n,k) = 1 - (365!/(365-k)!*365^n). This formula takes into account the number of possible combinations of k birthdays in a group of n people.

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