Hypergeometric Probability Distribution

Question

The game called Lotto sponsored by the Louisiana Lottery Commission pays its largest prize when a contestant matches all 5 of the 39 possible numbers. Assume there are 39 ping-pong balls each with a single number between 1 and 39. Any number appears only once, and the winning balls are selected without replacement.

 

Use the hypergeometric formula to find the probability of matching all 5 winning numbers. The lottery commission also pays if a contestant matches 3 or 4 of the 5 winning numbers. Hint: Divide the 39 numbers into two groups, winning numbers and nonwinning numbers.

Answer 1

Using standard variables,

N = total number of numbers = 39

n = number drawn (without replacement) = 5

K = total number of N which are winning = 5

k = number of winning sought = depends on whether we are matching 5, 4, or 3

 

In the first case, we have k = 5. Then we have 39!/34! ways of choosing 5 from the 39 (this becomes the denominator), and 5! ways of arranging the 5 winning numbers (this becomes the numerator), so 1/(39-choose-5) is the probability.

 

In the case of k = 3 or 4, we arrive at the standard formula for a hypergeometric distribution, which also works in the case above of k = 5.

 

(K-choose-k)(N-K-choose-n-k)

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           (N-choose-n)

 

To see this more intuitively, for k = 3 we have 5*4*3 ways to choose the three winning marbles, and 34*33 ways to choose the remaining two, non-winning marbles. Then we have 5-choose-3 ways to arrange what order those three winning marbles appear out of the five just drawn in sequence. Multiplying all this together gives the numerator. As in the case above, the denominator is 39*38*37*36*35.