You buy a lottery ticket to a lottery that costs 10 per tick

You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes.
Find your expected gain or loss.

Solution

Let X shows your expected loss or gain. So X can take values $500-$10 = $490, $100- $10 = $90, $25 - $10 = $15 and -$10.

When X=$490, you win $500 prize. Since there are 1 ticket out of hundred of $500 prize so

P(X=$490) = 1/ 100

Likewise

P(X = $90) = 2 / 100

P(X = $15) = 4 / 100

Out of 100, 100 - (1+2+4) = 93 tickets did not have any prize so

P(X = -$10) = 93 /100

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So expected value of your expected gain or loss is

E(X) = ($490) *(1/100) + ($90) *(2/100) + ($15) *(4/100) + (-$10) *(93/100) = -$2

That is your expected loss is -$2.