Answer:
The expectation for an event with outcomes:
{x₁, x₂, ..., xₙ}
Each one with probability:
{p₁, p₂, ..., pₙ}
Is:
Ev = x₁*p₁ + ... + xₙ*pₙ
There are 500 tickets sold.
1 of these, wins $3,000 (this is the event x₁)
4 of these, wins $100 (this is the event x₂)
2 of these, wins $75 (this is the event x₃)
The others do not have a prize.
So the probability of winning the $3000 is equal to the quotient between the number of tickets with that prize (1) and the total number of tickets (500)
p₁ = 1/500
Similarly, the probability of winning $100 will be:
p₂ = 4/500
And for the $75 prize:
p₃ = 2/500
Then the probability of not winning is:
p₄ = 493/500
Then the expected value for a single ticket is:
Ev = $0*493/500 + $75*2/500 + $100*4/500 + $3000*1/500
Ev = $7.1
If you take in account that you pay $50 for the ticket, the actual expectation should be:
E = $7.10 - $50 = -$42.90
