Question

The Powerball lottery commission chooses 5 white balls from a drum containing 69 balls marked with the numbers 1 through 69, and one red ball from a separate drum containing 26 balls. Assuming the jackpot for a certain drawing is $150 million, what is your expectation for the jackpot if you purchase 1 ticket of $2? Round to the nearest cent. Assume the jackpot is not spilt among the multiple winners

Answer 1

Probability = Favorable Outcomes / Total Outcomes

Please note ⁿC_x = n! / [(n-x)!*x!]

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Probability of winning the jackpot

(a) For the white Balls

Favorable outcomes : Choose 5 winning numbers = 1 way

Total Outcomes = ⁶⁹C₅ = 11238513

Therefore the probability of getting all 5 correct white balls = 1 / 11238513

(b) For the red Balls

Favorable outcomes : Choose 1 winning number = 1 way

Total Outcomes = ²⁹C₅ = 29

Therefore the probability of getting the correct red ball = 1 / 29

Therefore the probability of winning the jackpot = (1 / 11238513) * (1/29) = 1 / 325916877

Overall payout on winning = $150, 000 , 000 - 2 = $149, 999, 998

Therefore the expected Value on winning = 149,999,998 * (1 / 325916877) = 0.4602

Probability of losing = 1 - (1 / 325916877) = 325916876 / 325916877

Amount lost on losing = $-2

The expected value on losing = -2 * (325916876 / 325916877) = -1.9999

Overall Expected value = 0.4602 - 1.9999 = -1.5397      -1.54 dollars