There are 292.2 million total combinations in Powerball. The lottery sold 440.2

Question

There are 292.2 million total combinations in Powerball. The lottery sold 440.2 million tickets last drawing so you'd think there would have been a winner. How can this be? The answer is very sim-ple. I'd estimate that 200 million duplicate tickets were sold to unsuspecting players, with the vast majority being Quick Picks.

Let us make a mathematical model of this. We will number all the possible Powerball combinations by natural numbers 1 through N (for Powerball in its current form, N =292, 201,338). This way, we can think of the lottery as choosing a random number between1 and N.

We will assume that the lottery tickets are filled out randomly-with a uniform distribution-and independently. Suppose that K tickets are sold. We will assume that K > N.

QUESTION. In a drawing that sells K tickets, what is the expected number of duplicate tickets (i.e., ones that are picked two or more times)?

Explanation / Answer

Since the distribution is uniform.

if the Number of tickets are N it is expected that each ticket is printed atleast once.so expected number of duplicates is zero.

if the Number of tickets are N it is expected that each ticket is printed atleast twice.so expected number of duplicates is N.

so if K<N expected number of duplicates 0

if 2*N>K>N expected number of duplicates is K-N.

if 2*N>K>N expected number of duplicates is N

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