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.

Let us make a mathematical model. 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 between 1 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. What is the expected 'coverage" of the drawing in which K tickets are sold? In particular, for each number n between 1 and N, what is the probability that exactly n different combinations are chosen?

Explanation / Answer

Expected coverage = ticket solds per unique number = 440300000/292201338= 1.506
Probability of exactly n different combinations = KCn*(1/n)^k

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