Needing Math Geeks

I have a statistics problem I've been contemplating for a while, and I thought I'd share it.  I haven't figured out the "answer" yet, so I'm very interested to see what you all say.

Imagine that you are walking down the street and see a man on the sidewalk with a table set up.  He tells you that he has an offer for you.  If you give him a dollar, you can roll his standard, 6-sided die.  If you roll any number besides a number you pick, you get 2 dollars.  In fact, any amount you give him will be returned to you double if you win.  If you roll the number you picked, you lose your wager.  As an additional note, he says that you can play as many times as you want, but you must wager all of your earnings to continue.

Let's look at the math.  If you're satisfied that the man is honest and that the die is not tampered with, then you know you have a 5/6 chance to win.  That's awfully good odds considering that he's gonna double your money.  If you could play the game 6 times with just 1 dollar each time, you'd expect to walk away with 10 dollars (5 wins, 1 loss).  Unfortunately, you don't get that option.

Let's pretend you take the bet.  You choose number 1, and to your delight, you roll a 4.  He hands you 2 dollars and tells you to have a nice day.  Then you think, "I still have 5/6 odds to win, right?  I think I'll roll one more time." 
As is required for the game, you wager your 2 dollar earnings.  Sure enough, you pick 1 again and roll a 2.  You've now made 3 additional dollars on top of your 1 original.  You play again and increase to 8.  Again to 16.  Again, and you lose.  The guy keeps your money (15 in earnings and the 1 dollar you originally paid), and invites you to play again from the beginning.  What do you do?

Every time you walk up to the table, it makes statistical sense to take the bet.  5/6 odds are great for a double your money situation.  When you look at the game one set at a time, it always looks like a good idea, because you always have 5/6 chance to win.  But when you lose, you know you're gonna lose all your earnings.  And you also know that if you play long enough, you will lose.  No matter how well you are doing, you can't win forever.

Another look at the math - You have a 5/6 (83%) chance to win the first round.  If you want to calculate the chance of winning both the first and second rounds, though, it's 5/6 * 5/6 or 69.4%.  Three rounds in a row have only a 57.9% chance.  Here's the kicker.  If you just finished round tww, do you have a 57.9% chance to win the next round?  No.  You have an 83% chance to win.  Although you can foresee only 57.9% chance for three sucessive rounds, the closer you get to that third round, the more likely it will be that you're gonna make it, which makes sense as you already completed several rounds.  Here's a chart of your foreseeable chances to win to complete the round.

Round 1: 83.3%
Round 2: 69.4%
Round 3: 57.9%
Round 4: 48.2%
Round 5: 40.2%
Round 6: 33.5%
Round 7: 27.9%
Round 8: 23.3%
Round 9: 19.4%
Round 10: 16.2%

Okay.  Here are my 3 questions:

1 - Would you play this game if it were offered to you, and if so, how much would you initially wager?

2 - If you would play, how many rounds would you play, remembering that you must include all of your original wager and earnings with each new round?

3 - Who do you think would make more money?  The man at the table or the passers-by?

Here are some other considerations.  Say, for example, that you would play 3 rounds and wager 20 dollars.  This, of course, would give you a 57.9% chance of walking away with $80 after 3 rounds of play.  If, instead, you wanted to start with 5 dollars, you'd have to play 5 rounds, and you'd only have a 40.2% chance to walk away with $80.  What happens when you get to $80?  You still have an 83% chance to make it $160.  You were willing to hope for 57.9% odds.  Are you really going to pass up 83%?  Here the major problem I see: when you look several rounds into the future, you know you are likely to lose.  When you look at the next round, though, you know you are likely to win.  Would you resist going "just one more round"??  If you started with a nickel and managed to win 10 times in a row (a feat only accomplished by 16.2% of those that attempt it) and made 51 dollars, would you pass up that 83% chance to break 100?

Answer my questions and/or tell me what the math says to  you.  Do you think people should play until the chance drops below 50%?  Why can't they just mentally start over saying to themselves, "I've already won 4 rounds, but I still have a 69.4% chance of winning 2 more in a row"?  Shouldn't you look at each choice as it comes?  Of having an 83% chance?  But won't that ultimately lead to a loss?   :)

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