Question

The expected winnings are defined as the value of the jackpot multiplied by the probability of winning it. How large should the Powerball jackpot be in order for the expected winnings to exceed the ticket price ($2)?

The Powerball lottery drawing of January 13th, 2016 had the largest payoff in its history, of $1.6 billion dollars. The lottery draw consists of picking 5 white balls at random, from an urn of 69 balls, numbered from 1 to 69 (without replacement, and the order of the balls is not important). In addition, a “powerball” is picked at random from an urn of 26 red balls, numbered 1 through 26. For a $2 ticket you can choose the 5 winning white balls and the winning “powerball”. In this problem we will study the probability of winning this lottery. Assume that you purchase a $2 ticket.

Answer 1

5 balls can picked up without replacement from the urn of 69 balls in 69 * 68 * 67 * 66 * 65 ways = 1348621560 ways

The power ball can be picked up from the urn in 26 ways

So, the total number of ways to play the power ball lottery = 1348621560 * 26 = 35064160560

Probability of a win = 1/35064160560

Let the jackpot value be V

We want expected value > 2

[(-2)(35064160559/35064160560) + (V – 2) (1/35064160560)] > 2

On solving, we get V > 140256642240, that is V > $140.26 billion.