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A lottery has a grand prize of 400,000 dollars, five runner up prizes of 50,000 dollars each, nine third-place prizes of 10000 dollars each, and twenty-five consolation prizes of 1000 dollars each. If 2,000,000 tickets are sold for 1 dollar each and the probability of any one ticket winning is the same as that of any other ticket winning, find the expected return on a $1 ticket. (Round your answer to two decimal places.)

I tried $$(400,000\times \frac {1}{2,000,000}) + (50,000\times \frac {5}{2,000,000}) + (10,000\times \frac {9}{2,000,000}) + (1,000\times \frac{25}{2,000,000})$$

and so on with each of the other prizes, getting .3825 then round it to .38 which was wrong.

Please explain what I may have done wrong or show me what I would need to do instead. Thanks.

Kevin Diez
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2 Answers2

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I think your value of $.3825$ is correct for not taking into account the amount paid for the ticket.

So I think your expected value is $.3825-1=-\$0.6175$

paw88789
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  • Does the OP's wording, " expected return on a $1 ticket" support your interpretation? Probably you're right and the question is poorly phrased...) – DJohnM Oct 18 '17 at 19:29
  • @DJohnM: Considering that the OP did the math correctly (for the return not counting the ticket cost) but said the answer was 'wrong', I thought counting the cost of the ticket seemed like the most reasonable thing to do. – paw88789 Oct 18 '17 at 20:52
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You're missing the formula.

Let $p$ be the probability of winning, $C$ be the cost of a ticket, and $V$ be the value of the winnings. Then the expected value $E$ of a ticket, assuming one winner, would be approximately

$$E = p V - (1-p) C$$

Expected value of a bet = (probability of winning) x (winning amount) – (probability of losing) x (losing amount)

user16249
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