# Simple gambling problem

PBE6
Posers and Puzzles 01 Apr '08 18:12
1. PBE6
Bananarama
01 Apr '08 18:12
This one comes from the Old West, apparently.

A man with a hat shows you 3 cards, one completely gold, one completely silver, and one gold on one side and silver on the other. He puts them in his hat, and picks one at random. He then shows you one side of the card he picked, which happens to be silver. Now he says "I'll bet you even money that the other side of this card is silver too...whaddaya say, partner?"

Is this bet a fair one?
2. 01 Apr '08 18:432 edits
Yes, it's definitely not the completely gold card so it's either the gold/silver card or the silver/silver card. If it's the gold/silver card then the other side will be gold. If it's the silver/silver card the other side will be silver. Therefore the chance is 50/50.
3. 01 Apr '08 18:531 edit
The man with the hat already knows what colour the other side has, so he is placing the bet with "pre-knowledge". Accepting the bet would not be smart.
4. PBE6
Bananarama
01 Apr '08 19:03
Originally posted by heinzkat
The man with the hat already knows what colour the other side has, so he is placing the bet with "pre-knowledge". Accepting the bet would not be smart.
No, the dealer doesn't know what's on the other side of the card anymore than the contestant does. I should have made that more clear in the question.
5. PBE6
Bananarama
01 Apr '08 19:04
Yes, it's definitely not the completely gold card so it's either the gold/silver card or the silver/silver card. If it's the gold/silver card then the other side will be gold. If it's the silver/silver card the other side will be silver. Therefore the chance is 50/50.
Close, but no cigar. ðŸ˜‰
6. 01 Apr '08 19:311 edit
I wouldn't accept the bet.

Three possibilities:
(1) It is a fair bet. Then why did he give the offer if it's not good for him? A complete waste of time of his behalf.
(2) It is not a fair game:
(2a) He is the one losing in the long run, then he is crazy. Does he want to lose money or what?
(2b) He is the one winning in the long run, then he thinks I am the one who is crazy. He thinks he is smart.

The only probable alternative is 2a. Therefore I wouldn't pick up the bet. But I'd enjoy others to do it.

This decision is done without any use of probability or mathematics.

Edit: Should be 2(b) of course.
7. PBE6
Bananarama
01 Apr '08 19:36
Originally posted by FabianFnas
I wouldn't accept the bet.

Three possibilities:
(1) It is a fair bet. Then why did he give the offer if it's not good for him? A complete waste of time of his behalf.
(2) It is not a fair game:
(2a) He is the one losing in the long run, then he is crazy. Does he want to lose money or what?
(2b) He is the one winning in the long run, then he thinks ...[text shortened]... joy others to do it.

This decision is done without any use of probability or mathematics.
I think you meant 2(b), the bet is good for the dealer in the long run. Fair enough, although not exactly "rigorous".
8. 01 Apr '08 19:391 edit
Originally posted by PBE6
This one comes from the Old West, apparently.

A man with a hat shows you 3 cards, one completely gold, one completely silver, and one gold on one side and silver on the other. He puts them in his hat, and picks one at random. He then shows you one side of the card he picked, which happens to be silver. Now he says "I'll bet you even money that the other side of this card is silver too...whaddaya say, partner?"

Is this bet a fair one?
Not a fair bet. We should really be keeping track of sides, not cards. If, as supposed, you see a silver side, then that eliminates it from being the gold/gold card. So counting the remaining possible sides, there are two silvers, one gold -- each of which should be equally likely to be on the other side given what we know.

In other words, one who accepts the bet should on average lose 2 out of every 3 times.
9. 01 Apr '08 19:40
Originally posted by PBE6
I think you meant 2(b), the bet is good for the dealer in the long run. Fair enough, although not exactly "rigorous".
Quite right, I meant 2(b).

I've seen people who do the three card game at the street. It seems to me that they are stupid. And people win enorms sums of money. (Seems so, anyway. Rigged games of course.) When some innocent naïve spectator tries the same he lose a lot of cash.

I use the same reasoning as described above. If it is too good to be true, it's probably not true.
10. PBE6
Bananarama
01 Apr '08 19:46
Originally posted by LemonJello
Not a fair bet. We should really be keeping track of sides, not cards. If, as supposed, you see a silver side, then that eliminates it from being the gold/gold card. So counting the remaining possible sides, there are two silvers, one gold -- each of which should be equally likely to be on the other side given what we know.

In other words, one who accepts the bet should on average lose 2 out of every 3 times.
Correct!
11. 02 Apr '08 09:501 edit
Originally posted by PBE6
Correct!
Nice problem; there had to be something elementary that is easily missed.
12. 02 Apr '08 17:33
Originally posted by PBE6
Correct!
it comes down to this the odds are double for him to win because he is beting he drew the silver/silver card (1 assupmtion) while you are beting that he drew the silver/gold card and he drew it with the silver side twards you(2 assumptions!)
13. 02 Apr '08 18:08
The solution is quite simple.

if there is gold and silver in equal volume the gold is heavier; He is holding the card, so knows the weight: thus has information you don't have. How he makes money is very simple: he is quite framilliar with the weight.
14. PBE6
Bananarama
02 Apr '08 20:31
Originally posted by alexdino
it comes down to this the odds are double for him to win because he is beting he drew the silver/silver card (1 assupmtion) while you are beting that he drew the silver/gold card and he drew it with the silver side twards you(2 assumptions!)
Not quite.
15. PBE6
Bananarama
02 Apr '08 20:37
Originally posted by preachingforjesus
The solution is quite simple.

if there is gold and silver in equal volume the gold is heavier; He is holding the card, so knows the weight: thus has information you don't have. How he makes money is very simple: he is quite framilliar with the weight.
He doesn't need to know the weight of the cards to make money on this deal.