Probability?

Question

At a fair run by a local charity organization, it costs 50 cents to try one’s luck in drawing an ace from a deck of 52 playing cards. What is the expected profit per customer, if they pay $4 if and only if a person draws an ace?

Answer 1

The probability of winning is 4/52 = 1/13
The expected payout is (1/13)*$4 = $4/13 = about 31 cents.

Expected profit (for the charity) =
Cost of playing - expected payout
= 50¢ - 31¢
= 19¢

Answer 2

The probability that a customer draws an ace is 4/52=1/13. Thus the organizers are expected to pay 4/13 $per cusomer and earn 50 cents per customer. Thus there expected gain is 4/13$-50cents.

Answer 3

The "Expectation" is the weighted sum of all outcomes.

For example, if I give you a dollar per "point" on a card (i.e., three dolars if you pick a three), ten for any face card, then the calculation would look like

Amount * P =

ace: 1.00 * 4/52 = 0.076923...
two: 2.00 * 4/52 = 0.153846...
...
nine: 9.00 * 4/52 = 0.692308...
ten: 10.00 * 4/52 = 0.769231...
face:10.00 * 12/52 = 2.3076923

The "expectation" is the sum of all numbers on the right. In this case it is 6.54....

So, if I charge you less than $6.50 to try your luck, I will lose money over the long run. If I charge more than 6.55, I'll be expected to make a profit over the long run. How much profit?

(Fee - Expectation)*tries.

So, if I charge 7 bucks and have 1000 clients, I expect to make
1000*(7 - 6.54) = $460.

Your problem is much simpler (less calculations) because you have a lot of outcomes where the payout is 0.