What is the probability that two people selected from 30 have the same birthday?

Answer 1

To select the first birthday, the probability is 1/30. Having gotten that, the conditional probability that the next birthday would be the same is (1/30)x(1/29) and that is 1/870

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I believe the question has to be rephrased to "What is the probability that two people

in a group of 30 people share the same birthday?". Because in the way the question

is actually stated, "the probability that two persons selected randomly from a group of 30 have the same birthday", the event that "those two people would share their

birthday" is independent of the size of the population they were selected from.

In the case the actual question be "What is the probability that two people in a group

of 30 share the same birthday?, is given by the following expression that neglects

February 29 (of the leap), but gives very good approximation to the expression that

considers February 29 and is a simpler one. [It has to be mentioned that the analysis

leading to this expression considers birthdays a "random variable" where chances for

a persons birthday are the same for any day of the year]:

P(2 share bd out of n) = nC2 (1/365) Π1n-1[1-(i-1)/365]

for n = 30, P(2 share bd out of 30) = 30C2 (1/365) Π1 29 [1-(i-1)/365] = 435∙(1/365)∙

[1-1/365]∙[1-2/365]∙[1-3/365]∙ ∙∙∙ ∙[1-28/365] = 0.380215577... ≈ 0.380 ≈ 38.0%

For the construction of the expression to calculate the probability of any two people sharing a birthday in a group of n people considering Feb 29 of the leap year see the

question "What is the probability that in a room of 8 people 2 have the same birthday?"