Sharing Birthdays

I was introduced to this challenge as a high school student. It was one of the most exciting moments of my student life! So here's the challenge..

In a class of 40 students, what is the probability that 2 students share the same birthday?

Very less! Don't you think so? 40 students and 365 days in a year and their birthdays matching... very very remote. However, against our intuition the probability is as high as 0.89!

The calculation is really simple. The probability that the birthdays of 2 students matches is (the second student has one chance in 365)

Now the probability that the birthdays of the 2 students doesn't match is

The probability that a third student doesn't have the same birthday as the first 2 students is

The probability that the 3 students don't share the same birthday is

(the probability of the birthday of the second student doesn't match with the first student AND the probability of the birthday of the third student not matching with the first and second students – product of both probabilities).

On similar lines, the probability that no 2 students have the same birthday out of a class of 40 is

and hence, the probability that 2 students have the same birthday out of a class of 40 is the complementary

Thats how the 89% chance that 2 students of a class have the same birthday. Observe how the probability increases steeply. Note that the Y axis in the graph is a logarithmic scale.

Related reading - 'Math Chambers' by Alfred S Posamentier