the smallest number of people in a room where the probability of two of them having the same birthday is at least fifty percent
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The smallest class where the chance of finding two people with the same birthday is 50% or more is... a class of 23 people. (The probability is about 50.73%.)
The smallest number of people in a room where the probability of two of them having the same birthday is at least 50 percent.
To understand why a class of 23 people is the smallest where the probability of two people having the same birthday is at least 50%, we can use a concept called the Birthday Paradox.
The Birthday Paradox states that in a group of people, the chances of two people sharing the same birthday are higher than most people would intuitively think. To calculate this probability, we can use the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.
To solve this problem, we need to find the smallest number of people in a room where the probability of two people having different birthdays is less than or equal to 50%, which means the probability of two people having the same birthday is at least 50%.
Let's calculate the probability of two people having different birthdays in a class of n people. The probability of the first person having a different birthday than the second person is 365/365 (since there are no restrictions on the first person's birthday). The probability of the third person having a different birthday than the first two is (364/365), and so on.
So, the probability of n people having different birthdays is (365/365) * (364/365) * (363/365) * ... * ((365-n+1)/365).
We want to find the smallest value of n for which this probability is less than or equal to 0.5 (50%).
By trying different values of n, we can find that a class of 23 people is the first where the probability of two people having different birthdays is less than or equal to 0.5. The probability of two people having the same birthday in a class of 23 people is approximately 50.73%.
Therefore, the smallest number of people in a room where the probability of two of them having the same birthday is at least 50% is 23.