Birthday!

You have $n$ people at a birthday party. You want to determine the probability that any two of them do not have their birthday on the same day of the year. Let $p(n)$ denote that in a set of $n$ people chosen at random, at least two people share the same birthday. We have:

$p(n) = 1 - \frac{365 \times 364 \times ... \times (366 - n)}{(365)^n}$ = $1 - \frac{365!}{(365 - n)! 365^n}$

Given $n$ people at a party, we want to determine: what is the probability that any two of them do not have their birthday on the same day of the year? This is simply the complement of $p(n)$ . Thus, the probability is:

$1 - p(n)$ = $1$ - $(1 - \frac{365!}{(365 - n)! 365^n})$

In this case, we have $n = 34$ and thus we have that if you have 34 people at a party, the probability of any two of them not having their birthdays on the same day of the year is:

$1 - p(34)$ = $1$ - $(1 - \frac{365!}{(365 - 34)! 365^{34}})$ $\approx$ 20.46

The first person can have his birthday on any of the 365 days. So the second person's birthday must be on any other day except the day on which first person was born i.e 364 days. It goes on like this and the last person i.e 34th person's b'day must be on any of the remaining 332 days. Therefore the probability is 364/365 363/365 362/365....... 333/365 332/365

https://brilliant.org/wiki/birthday-paradox/ It is worth noting that this is a pretty big calculation from the equation we were given, so I will be going through 1 possible programming solution My python (I used python for this fairly simple and one of calculation as python is simple) code is this:

#we will look at the probability that an individual will not share a birthday with all previous individuals, and account for it on our probability, then repeat the 
process, until we run out of individuals
p=100#you start at 100% as there is no individual to share his/her birthday with
for i in range(34):#this is just accounting for all the individuals
    p=p*((365-i)/365)#here I am just multiplying p by the probability the (i+1)th person has a new birthday, which is just the number of days left to choose from 
(i.e. 365-the number of birthdays already chosen=365-the number of individuals from before=365-i) divided by the number of days in a year (365)
print(p)#just displaying the result