Question

How many people do we need to have in a room to make it that the probability of two people in the room will have the same birthday is greater than ½? (Note: Here we consider just the day and month, not the year.)

Answer 1

here let number of people required =a

total number of ways to select 2 people from group of a =(_aC₂) =a(a-1)/2

probability that two random person have same birthday p=1/365

expected number of pair of people with same birthday =np =(a*(a-1)/2)*(1/365) =a(a-1)/730

therefore from Poisson approximation:

P(at least one pair have same birthday) =P(X>=1) =1-P(X=0) =1-e^-a(a-1)/730 >=1/2

e^-a(a-1)/730 <=1/2

taking log on both side:

a²-a >=-730 ln(0.5)

a² -a +505.96 >=0

solving above quadratic equation:

a = (1 -/+ √(1²-4*1*505.96))/(2*1) = -22.488 or 22.488

since a can be positive only

a >=22.488

or a =23

therefore minimum number of people required =23