a First assume the probability of being born on every day is

(a) First, assume the probability of being born on every day is equal (n = 366 days). Compute the probability of getting at least one birthday match in a class of k = 40 students. (b) We can study the effect of unbalance using a simplified model, the (un)biased coin. Call the probability of heads on a coin p. What is the probability of getting the a match in two throws of the coin? How does this probability change when p not equals to 0.5.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where     
k = number of successes in trials        
n = is the number of independent trials        
p = probability of success on each trial
a)
We will take the Probability of a Birthday on Specfic day = 1/366 = 0.0027

We gonna find the matched day atleast for one birthday
P( X < 1)   = P(X=0)       
= ( 40   0 ) * 0.0027^0 * ( 1- 0.0027 ) ^40
= 0.8975

P( X > = 1 ) = 1 - P( X < 1) = 0.1025