what is the expect profit?

Question

A game is played in a way that you draw three cards from a standard deck of 52 cards. If all the cards are queens, then you win 100,000 pesos. Otherwise, you pay 500 pesos. If you play this game repetitively, will you expect to profit from playing the game?

Answer 1

As there are 4 queens out of 52 cards in the deck, the number of hands in which you win is C(4,3), and the number of possible hands is C(52,3).

Thus, the probability of you winning is 4*3*2/3!/(52*51*50/3!) = 4 * 3 * 2/(52*51*50) = 1/(13*17*25) =1/5525, or the odds of you winning are 1:5524 (I mention odds because, the way the problem is stated, you don't pay the 500 pesos unless you lose). 100000:500 = 2000:1. The winning payout needs to be 5524 times the losing fee for breakeven but it is less, so on average, you lose.

You can also determine the expected payoff. 1/5525 * 100000 + 5524/5525 * -500 = -106480/221 = -481.81 pesos. (Note the presence of 5524/5525; this is because you only pay the 500 pesos if you lose).