Very simple expectation problem

[asdf1234]

Gambling game, a man is paid $3 if he draws a J or Q, $5 if he draws K or A from ordinary deck. Any other card and he loses. How much should he pay to play if the game is fair? The only reason I ask is that I think a book has the answer wrong.

[Copernicus]

$1.23

[asdf1234]

Okay, that's what I thought, the book had $1.78, forgetting to take into account that you pay when you win I think.

[BruceZ]

$1.78 is what he should pay if he gets his money back when he wins, in addition to the money he wins.

EV = 8/52 * 5 + 8/52 * 3 - 36/52 * x = 0

x = 1.78

If he doesn't get his money back, then he should only pay $1.23.

EV = 8/52 *(5-x) + 8/52 * (3-x) - 36/52 * x = 0

x = 1.23

[Copernicus]

the problem is posed as you are "paid 3 and paid 5" not you "win 3 and win 5", so unless the problem is worded poorly $1.23 is a better answer than $1.78.

Its like the old ploy of saying they are giving you 3 for 1, instead of 3 to 1 odds. In the latter the 3 is in addition to your stake, in the former it includes it.

[asdf1234]

The book specifically says he is paid $3 or $5 if he wins, so according to the wording of the question, the answer needs to be $1.23. If they actually meant he WINS (i.e. profits) $3 or $5, then they worded it very very very poorly. I'll probably have to give credit for either answer.

[Copernicus]

Give them credit for either answer, or challenge them and require two different answers along with the circumstances that make each one correct.

[Bozeman]

I think it is only a semantic issue. "paid" could be the amount won, or the amount won + amount returned.

Still, it would be nice if the phrasing were not ambiguous.