Different Probablities Same Situation?

[dkgolfer16]

Johnny and Sally sit at a table in their dining room. Sally tells Johnny to leave the room while she prepares a game. Sally randomly selects three cards from a regular deck of cards (half black, half red) and places them face down on the table. She yells at Johnny to reenter and tells him that he gets to flip two of the three cards over. If two are of the same color, Johnny wins $2. If they are different colors, Sally wins $1.

Johnny's viewpoint: It's a great deal because I have a 50% chance of winning $2 and 50% chance of losing $1. Why does he think this? The first card color he flips over is of no difference. The second card he flips has a 50% chance of being either red or black, thus 50% chance of matching the first color.

Sally's viewpoint: It's a great deal because Johnny only has 33% chance of winning so theoretically I should pay him $3 if he wins. Why does she think this? Since three cards exist, the odds of flipping over two of the same color are 1 out of 3.

Question: Who is right? Who will win? Are they somehow both right?

Thanks for the help.

 

[robert Ihnot]

If you reduced this to dealing two cards and asking the odds, how does that differ from chosing two cards at random from a deck of 52?

 

[regor60]

Johnny's right. By his logic. His expected value is 50 cents. Her logic is incomplete. If you focus on three cards, there are twelve combinations, 6 of which are of same color, therefore 50%. Even if he had a 1/3 chance of winning, it still wouldn't be a great deal, because her expected return is 0 with his $2 winning potential. Paying him $3 would be to her disadvantage, her expected return would be negative 33 cents

 

[dkgolfer16]

That's what I thought. Johnny's right. But this link was posted to explain bell's inequality. Maybe I'm misunderstanding the example but it says the odds are on Sally's side. See "Is this game fair to you?" heading at the following address:

http://ilja-schmelzer.de/realism/game.php .

Correct me if I'm missing something.

 

[CRGreathouse]

Johnny's essentially right -- though his expected winnings is more like 47 cents (8/17) because the deck isn't infinite.

Correct me if I'm missing something.

In the link there are three preselected cards, two of the same color. In your example the three cards may be all the same color. Text comparisons:
"Sally randomly selects three cards from a regular deck of cards"
"I put three cards of my choice on the table"

 

[robert Ihnot]

dkgolfer16: Sally randomly selects three cards from a regular deck of cards

That's not what the example says, it says: "I put three cards of my choice on the table so that you cannot see their color."

 

[dkgolfer16]

Got it thanks