SOLUTION: A lottery offers on $1000 prize, one $500 prize and five $100 prizes. Seven hundred tickets are sold at $4 each. Find the expected value of a ticket if a person buys one ticket.

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Question 1142813: A lottery offers on $1000 prize, one $500 prize and five $100 prizes. Seven hundred tickets are sold at $4 each. Find the expected value of a ticket if a person buys one ticket.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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A gamer either wins $1000 prize with the probability  1%2F700,  

            or wins  $500 prize with the probability 1%2F700,  

            or wins any one of five  $100 prizes with the probability 1%2F700,

            or wins NOTHING (= any one of the remaining 700-1-1-5=693 void prizes) with the probability 1%2F700.


Thus the mathematical expectation of winning for 1 single ticket is


    1000%2F700+%2B+500%2F700+%2B+%285%2A100%29%2F700+%2B+%280%2A693%29%2F700 = %281000+%2B+500+%2B+500%29%2F700 = 2000%2F700 = 20%2F7 dollars.


From it, subtract the price $4 of the ticket, and you will get the expected value of the ticket


     20%2F7-4 = %2820-28%29%2F7 = -8%2F7 dollars = -11%2F7 dollars.    ANSWER


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solution by tutor @ikleyn shows the standard method for finding expected value, based on the definition.

For this particular problem (and many similar ones), the expected value can be found much more easily.

(1) The total ticket sales are 700($4) = $2800.
(2) The total payouts are $1000 + $500 + 5($100) = $2000.

The total gain by the ticket buyers is $2000-$2800 = -$800; the expected value of each one of the 700 tickets is -$800/700 = -$8/7.