Question 1131374
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I'm not sure what the meaning is of the phrase "expected value of the lottery".  I will assume you mean the expected value of one ticket....<br>
1 first prize; value $400-$6 = $394 (prize minus cost of ticket)
2 second prizes; value each $200-$6 = $194
5 third prizes; value each $150-$6 = $144
992 no prize; value each -$6<br>
The expected value of a ticket is then<br>
{{{(1/1000)*394+(2/1000)*194+(5/1000)*144+(992/1000)*(-6) = (394+388+720-5952)/1000 = -4450/1000 = -4.45}}}<br>
The expected value of one ticket is -$4.45.<br>
While that is the classical way to calculate the expected value, based on the definition, there is a much easier path to the answer.<br>
The total cost of the 1000 tickets is $6000, the total payout is $400+$400+$750 = $1550.  The expected value of each ticket is<br>
{{{(1550-6000)/1000 = -4450/1000 = -4.45}}}