20071024, 17:44  #1 
May 2004
New York City
2×2,099 Posts 
Coin Toss Game
A casino opens a new game using a fair (5050) coin toss.
The player pays (bets) one chip to play. The coin is then tossed repeatedly until one TAILS occurs. The player gets paid one chip for each HEADS that occurs before that first TAILS. So for example if TAILS happens on the first coin toss, the player just loses his one chip bet; if one HEADS precedes the TAILS, the player comes out even; if two HEADS precede the TAILS, the player comes out one chip ahead; and so on. The question is: is this a fair game, or does the house (or player) have a monetary advantage (and if so, how much)? Now in the corner of the casino is the high stakes version of the game, with different payment rules. Here, the player pays M dollars to play, but gets paid 2^N dollars if N HEADS occur before the first TAILS. The question here is: for what value of M is this a fair game for the player and the house? 
20071024, 19:36  #2  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
17·251 Posts 
Quote:
With the second, M = 2 seems to get the closest to being fair, but there does seem to be a slight favor in the player's direction. I wrote a program to calculate the numbers. 

20071024, 22:57  #3 
"Lucan"
Dec 2006
England
6,451 Posts 
Expected return is:
1*(1/2)^2+2*(1/2)^3+3*(1/2)^4+........ This is the sum of GPs: (1/2)*[(1/2)+(1/2)^2+(1/2)^3)+....]+ (1/2)^2*[(1/2)+(1/2)^2+(1/2)^3)+....]+ (1/2)^3*[(1/2)+(1/2)^2+(1/2)^3)+....]+ (1/2)^4*[(1/2)+(1/2)^2+(1/2)^3)+....]+ .... Now [(1/2)+(1/2)^2+(1/2)^3)+....]=1, so the expected return is 1 I agree the game is fair. The second game has something to do with St Petersburg IIRC David Last fiddled with by davieddy on 20071024 at 23:52 Reason: I could have omitted brackets round (1/2) with the advantage of brevity 
20071025, 00:12  #4 
May 2003
2×761 Posts 
The first game is "fair" but ultimately loses for the casino if players are smart. (Yes, the expected return is what you pay, but eventually with 100% probability, there is some point at which the casino will owe you more than you paid. So you stop at that point. Of course, this assume you have an infinite amount of money, and that you keep playing when you are VERY down. Although, with a fair coin, it is unlikely that you will have to get down too much before you are up at least once.)

20071025, 00:41  #5 
"Lucan"
Dec 2006
England
6,451 Posts 
ZetaFlux I think you need to read the original problem
more carefully. That said your observations have some relevance to the second problem. In the first case, quitting before a tail turns up would be a silly move (unless the bouncers were getting trigger happy) Last fiddled with by davieddy on 20071025 at 00:42 
20071025, 02:15  #6 
Aug 2002
Ann Arbor, MI
1B1_{16} Posts 
I think the second version would be more fun if he got paid (2^N)/N dollars if it took N tosses to get a tail...

20071025, 03:20  #7  
Jun 2003
7×167 Posts 
Quote:


20071025, 09:08  #8 
"Lucan"
Dec 2006
England
6,451 Posts 

20071025, 09:33  #9  
"Lucan"
Dec 2006
England
6,451 Posts 
Quote:
May I recommend "Gamblers Anonymous"? David 

20071025, 23:09  #10 
May 2003
2×761 Posts 
I do not gamble. So don't worry about me going out and spending all my money to win it back. ;)

20071026, 11:17  #11 
"Lucan"
Dec 2006
England
6451_{10} Posts 

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