Expected Value in Games

Question

Imagine you are doing research for a term paper in your English class and you have chosen the topic to be “The Benefits and Costs of My State's Lottery”. You find the following information for your state’s lottery:

              Lottery summary
Prize                              Probability
Jackpot                       1 in 200,000,000
$16,000                        1 in 180,000
$600                               1 in 15,000
$5                                     1 in 170
$2                                       1 in 80

You know that it costs $2 to purchase the lottery ticket in order to play the game. What would you conclude in your term paper regarding the expected value of the game (after purchasing the ticket), if the jackpot were to be set at $190,000,000?

$ (Round to the nearest cent.)

Answer 1

You can calculate the expected pay-out from gambling by multiplying the reward by the probability of winning that reward.You have a chance to earn multiple different rewards in this case, so we can make the total expected payout the sum of the expected payout for each individual prize.

Jackpot: ($190,000,000)*(1/200,000,000) = $0.95

2nd prize: ($16,000)(1/180,000) = $0.09

3rd prize: ($600)(1/15,000) = $0.04

4th prize: ($5)(1/170) = $0.03

5th prize: ($2)(1/80) = $0.03

Expected payout = $0.95 + $0.09 + $0.04 + $0.03 + $0.03 = $1.14

So you can expect to win $1.14 per ticket, but it costs $2.00 to buy one. The net value will be negative: $1.14 - $2.00 = -$0.86

This is how lotteries and chance-based games at casinos make money. A few individuals may win big, but on average the game costs more than it pays out.