IAI November 2010 Examination Q3

Can we use combinations for solving this problem? If yes, please mention the solution.

 

(a)
Since p = 1/365 and q = 364/365
The probability of occurring one person's birthday is [1/365]
The probability of occurring two person's birthday is [1/365] * [1/365]
In a complete year the probability of occurring two person's birthday is
[ [1/365] * [1/365] * 365 ] = 0.002740

(b)
Say X is the number of birthdays, and hence X ~ Bin (3 , 1/365)
In a complete year occurring of at least two person birthdays is
P[ X => 2] * 365 = [ P [ X = 2 ] + P [ X = 3] ] * 365
= [3C2 * (1/365)^2 * (364/365)^(3 - 2) + 3C3 * (1/365)^3 * (364/365)^0 ]
= 0.008204

Similarly

Say X is the number of birthdays, and hence X ~ Bin (4 , 1/365)
In a complete year occurring of at least two person birthdays is
P[ X => 2] * 365 = [ P [ X = 2 ] + P [ X = 3] + P [ X = 4] ] * 365
= [4C2 * (1/365)^2 * (364/365)^(4 - 2) + 4C3 * (1/365)^3 * (364/365)^1 + 4C4 * (1/365)^4 * (364/365)^0 ] * 365
= 0.016356

(c)
Say X is the number of birthdays, and hence X ~ Bin (15 , 1/365)
In a complete year occurring of at least two person birthdays is
P[ X => 2] * 365 = [ 1 - P[ X < 2] ] * 365 = 0.25

 

Why are we multiplying each probability with 365?

 

Actually in the question it was asked us to get the probability for 365 days.

 

Sorry but I didn't get you

 

In the question it was asked us to get the probability as a whole i.e., for 365 days. That's why we have multiplied it with 365 days.