Same Birthday

Rohit is the best friend of Shwetha. Once during Shwetha’s birthday party, Rohit went on to the stage and asked the people who had assembled there to bet. According to the bet, in that group of about 50 people, if there were at least two people with same birthday, then every person of the group would give Rohit Rs.500. If no two people with the same birthday were found, then Rohit would give Rs.25,000 to the entire group. Shwetha confirmed that the details of none of the people were known to Rohit. Then all people agreed to the bet. After revealing all of the people’s birthdays, it was found that indeed two people were found with the same birthday. Rohit collected Rs.500 from each person and gave it to Shwetha as a gift. As Rohit was very sure of winning the bet, his probability must be greater than 50%, but exactly how much? Curious Shwetha calculated the probability. She was amazed.

Can you find the chance of Rohit winning the bet?


Hint 1

Find the probability of all 50 people having different birthdays.


Hint 2

If Mathematics is difficult for you, try programming.


Answer

The chance or probability of Rohit winning the bet was 97.03%.


Solution

The problem statement has the words like probability and chance. So this must surely be a problem in probability. Ok. The problem seems interesting. The problem is simple but it is made big so as to create an illusion of the problem being difficult to solve.

Let us simplify the problem. The problem asks us to find the chance or probability that at least 2 people have the same birthday (i.e. their birthdays fall on the same day of the week) among 50 people.

We know from probability that,
(Probability that at least 2 people among n people have same birthday) = 1 – (Probability that all n people have different birthdays)

Let us denote


The problem asks us to find S₅₀ and it is very difficult to find it without using the above formula.

Now let us find D_n

We assume that there are only 365 days in a year and that all birthdays are equally probable, which are not quite met in reality.

The first of the n people has of course some birthday with probability 365/365 = 1. Then, if the second person is to have a different birthday, it must occur on one of the other 364 days. Therefore, the probability that the third person has a birthday different from the first two is 363/365. Finally, the probability that the nth person has a birthday different from the others is (365-n+1)/365. We therefore have








Taking logarithm on both sides


We know a formula from Calculus




Expanding and simplifying, we have


Substituting n = 50 and considering only the significant terms, we have







Therefore, there was a 97% chance or probability of Rohit winning the bet and only 3% probability of him losing the bet. Even after proving the result, it is an unbelievable fact. Rohit knew all about this. So he bet. Awesome!!!

There may be two types of questions
a) To find S_n when n is given. Ex: the above problem.
b) To find n when S_n is given. Ex: find the minimum number of people required to make the probability that at least 2 people among n people have same birthday greater than 50%.

The latter problem is like this.
Find minimum i such that



Or Find maximum i such that



For small n compared to 365 i.e. for n < 30, in the formula, the second and higher degree terms are neglected and hence a good approximation is




Here substituting D_i = 0.5 and simplifying we get n = 23

Find out for yourself these facts

n	10	20	23	30	40	50	60	70	80
Sn in %	11.69	41.14	50.72	70.63	89.12	97.03	99.41	99.92	99.99

All the aforementioned calculations are fine, right, but who on earth will know when to take logarithms and when to apply those unknown formulae from Calculus??? Ok. Ok. If you are fed up by all these tedious proofs and creepy calculations, then you can do one thing. Very simple. Ask the computer to solve your problem by calculating the stuffs. Simple







From a program, we can find both




Click here to download the source code of the program.