Example 48

EXAMPLE 48 Florida Lotto

You can win the jackpot in the Florida Lotto by correctly choosing all 6 winning numbers out of the numbers 1–53.

What is the number of ways of winning the jackpot by choosing all 6 winning numbers?
What is the number of outcomes in this sample space?
If you buy a single ticket for $1, what is your probability of winning the jackpot?
If you mortgage your house and buy 500,000 tickets, what is your probability of winning the jackpot (assuming that all the tickets are different)?

Solution

The number of ways of winning the jackpot by correctly choosing all 6 of the winning numbers and none of the losing numbers is
$N (Jackpot) =_{6} C_{6} \cdot_{47} C_{0} = 1 \cdot 1 = 1$

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The size of the sample space is

$\begin{array}{l} N (S) & = {}_{53}C = \frac{53!}{6! (53 - 6)!} = \frac{53 \cdot 52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47!}{6! \cdot 47!} \\ = \frac{16, 529, 385, 600}{720} = 22, 957, 480 \end{array}$
Therefore, if you buy a single ticket for $1, your probability of winning the jackpot is given by

$P (Jackpot) = \frac{1}{22, 957, 480} \approx 0.00000004356$
If you buy 500,000 tickets and they are all unique, then your probability of winning becomes

$P (Jackpot) = \frac{500, 000}{22, 957, 480} \approx 0.02178$

This is because the unique tickets are mutually exclusive, and the Addition Rule for Mutually Exclusive Events allows us to add the probabilities of the 500,000 tickets.After mortgaging your $500,000 house and buying lottery tickets with the proceeds, there is a better than 97% probability that you will not win the lottery.