Question

(probablity): given 3 groups of people: first group with 20 people, second group with 30 people, third group with 45 people.

a)for each group: calculate the probability that at least 2 people will have a birthday in the same day

b)write a program that generates random dates and run it on each group. how are the results obtained from running the program in comparison to the probabilities calculated for each group?

Answer 1

The probability for two people in a group to have the same birthday - formula 1-(365!/{(365-no. of people in the group)! * 365^no. of people in the group})

Group of 20 - 0.411 i.e approximately 41.1%

Group of 30 - 0.704 i.e approximately 70.4%

Group of 45 - 0.941 i.e approximately 94.1%

Program to generate random numbers for each member of the group (the numbers indicate the days out of 365 days)

import random

def Rand(start, end, num):
   res = []

   for j in range(num):
       res.append(random.randint(start, end))

   return res

num = 45 #indicates number of members in the group(change this accordingly)
start =1 #start day of an year
end = 365 #end day of an year
print(Rand(start, end, num))

Here, we can see that 4 pairs of people i.e 8 members shared their birthday with someone else.

Here, we can see that for a group of 20 only 1 pair that is 2 people out of 20 share their birthdays with someone else.

Here, we can see that 2 pairs share their birthday that is 4 people among 30 share their birthdays with someone else

As the probability increased from a group of 20 to a group of 30 to a group of 45, the number of members sharing their birthdays with someone else has increased from a group of 20 to a group of 30 to a group of 45.

Observation :

PROBABILITY	PROGRAM
41.1 %	2 PEOPLE OUT OF 20
70.4 %	4 PEOPLE OUT OF 30
94.1 %	8 PEOPLE OUT OF 45