statistics expected gain or loss

Question

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 $445 prize, two $95 prizes, and four $20 prizes. Find your expected gain or loss. (Round your answer to two decimal places.)

Answer 1

if you bought all 100 tickets you'd win 445+2(95) + 4(20) = 445+190+80 = 445+270 = $715

you paid 10(100) = $1,000 to win $715 = a net loss of $285

$285/100 = $2.85 expected loss per ticket

odds are you'd lose $10 on 1 ticket or on 93 of the tickets

you'd win $445 on 1 ticket, $95 on 2 tickets, $20 on 4 tickets

you'd lose on 93 tickets and win on 1+2+4= 7 tickets

weight each outcome by the number of tickets, then divide by the total weights of 100:

93(-10) + (1)(445-10) + (2)(95-10) + (4)(20-10) = -930 + 435+ 170+ 40 = -930+645 = -285

-285/100

= an expected loss of $2.85 per ticekt

the "expected loss" is not an actual outcome. buy one ticket and you won't lose $2.85. It's an average, a weighted average of all outcomes. It's like a typical average family has 2.1 children. Not one couple actually has 2.1 children, but it's the "expected" number of children for any randomly chosen couple.