perplexus.info :: Games : 3 cards

3 cards (Posted on 2002-05-11)

A dealer offers to you to play a game. He shows you three two-sided cards: one with both sides red, one red and black and the other black and black. He puts them in a hat, and randomly (no tricks here) takes out a card and puts it on the table.

You both see only one side of the card. At this point he says that if the bottom side is the same as the top, he will take your money. If the other side is different, you double it. He explains that by now one of the cards is ruled out - if you're seeing red, the card cannot be a double black card, and vise versa - so you have a 50/50 chance of winning.

Is this a fair game? Why or why not?

Submitted by levik

Rating: 3.5385 (13 votes)

Solution: (Hide)

The dealer is of course trying to trick you. His logic that one of the two possible cards is a winner is flawed. Let's say the card is showing a red face. The dealer would have you believe that it is either a red-black card or a red-red card, and therefore your chance of winning is 50/50.
But in reality, there are three possibilities:
it's the red-black card with the red side up
it's the red-red card with the first red side up it's the red-red card with the other red side up
As you can see, only one of these is a winning combination, so your odds of winning are 1/3. Therefore, the payout is not enough to make the game fair.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Odds	RandyOrton	2004-10-21 15:14:57
seems simple to me	Dan Porter	2004-01-20 11:07:00
A-priori probability	Andy	2003-11-20 11:22:59
I agree with the solution	S	2003-10-30 23:24:03
idea	jen	2003-10-29 18:11:04
re: Clarified explanation (solution)	Gamer	2003-07-14 17:08:00
No Subject	molly	2003-06-11 17:53:58
Clarified explanation (solution)	Ryan	2003-05-07 07:22:41
no fair	Mel	2002-08-23 03:44:27
re(3): Simply stated (solution)	levik	2002-07-16 08:40:34
re(2): Simply stated (solution)	TomM	2002-07-16 00:56:54
re: Simply stated (solution)	levik	2002-07-15 18:22:02
Simply stated (solution)	Henry	2002-07-15 09:58:05
solution?	marty	2002-05-22 22:44:06
bleh	Tom Bayer	2002-05-13 21:36:45
1/3	Michael	2002-05-12 19:18:31
The choices	David	2002-05-12 09:33:08
My thoughts	TomM	2002-05-11 07:15:40

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