Find the probability that of 25 randomly selected students, no two share the same birthday.

Solution:

Probability of an event E= favorable cases to E / total number of events.

Probability of an event = n(E)/n(s)

Given that, there are 25 randomly selected students.

Let us assume that there are 365 days in a year.

As anyone can have his birthday out of 365 days, the total number of ways = (365)²⁵

To give each one a different birthday, we can let everyone choose a birthday as his/her turn comes.

The first student will have 365 choices, and the second will have 364 choices, the third will have 363 choices, and so on...,

The last student will have 341 choices.

⇒ All different birthday ways is found using permutations = nP\(_r\) = 365P\(_{25}\)

Therefore, P(no two share the same birthday) = [365P\(_{25}\)/(365)²⁵]

=0.4313

Find the probability that of 25 randomly selected students, no two share the same birthday.

Summary: 

The probability that of 25 randomly selected students, no two share the same birthday is =0.4313