Who has the advantage? Dice game puzzlePrev TopicNext Topic

• db101

  

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  Jan 11, 2023, 9:15 pm

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  This was a puzzle from my kid's advanced math class. Stop me if you've heard this one before.

  There's two people, Alice and Brad, who each have a six sided die. Alice's has 3 red faces and 3 white faces. Brad's has 2 red faces and 4 white faces.  Alice proposes a game where they take turns rolling and whoever gets red first wins. Brad says it's not fair because Alice's die has more red faces. So Alice says Brad can go first to make up for it. Which one is more likely to win the game?
• cottoneyedjoe

  

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  Jan 12, 2023, 10:50 pm

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  In response to db101

  Quote: Originally posted by db101 on Jan 11, 2023

  This was a puzzle from my kid's advanced math class. Stop me if you've heard this one before.

  There's two people, Alice and Brad, who each have a six sided die. Alice's has 3 red faces and 3 white faces. Brad's has 2 red faces and 4 white faces.  Alice proposes a game where they take turns rolling and whoever gets red first wins. Brad says it's not fair because Alice's die has more red faces. So Alice says Brad can go first to make up for it. Which one is more likely to win the game?

  I've heard it before, but I won't stop you. They both have a 50% chance of winning, so it's a fair game. 

  One way of solving it is like this: Brad makes the first roll and has a 1/3 chance of winning. That means there's a 2/3 chance that the turn passes to Alice for the second roll. Her chance of winning is therefore (2/3) x (1/2) = 1/3, same as Brad. And the remaining 1/3 of the time nobody wins. If nobody wins, the game resumes as if the first two rolls never happened. Neither player has an advantage over the other. (There is only an infinitesimally small chance the game lasts for infinity rolls.)
• db101

  

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  Jan 14, 2023, 5:13 pm

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  In response to cottoneyedjoe

  Quote: Originally posted by cottoneyedjoe on Jan 12, 2023

  I've heard it before, but I won't stop you. They both have a 50% chance of winning, so it's a fair game. 

  One way of solving it is like this: Brad makes the first roll and has a 1/3 chance of winning. That means there's a 2/3 chance that the turn passes to Alice for the second roll. Her chance of winning is therefore (2/3) x (1/2) = 1/3, same as Brad. And the remaining 1/3 of the time nobody wins. If nobody wins, the game resumes as if the first two rolls never happened. Neither player has an advantage over the other. (There is only an infinitesimally small chance the game lasts for infinity rolls.)

  Thats a good solution. The textbook solution was summing a geometric series that converges to 1/2.
• eddessaknight

  

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  Jan 31, 2023, 9:58 pm

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  This informative  Dice odds contribution credit  by,  our professional gaming LV member

  COIN TOSS:

  • "For a real good lesson in odds consider how the odds of a pair of dice (each die with six pips on it)  rollings 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12  come into being. 

    There's one and only one way to roll a 2, a one and a one. Two ways to roll a 3, a one and a two or a two and a one. Go from there.  I'll help you out, 4, 5, ,and 6 are the same as 8,l9,and 10 for odds purposes.Six ways to roll a 7. 

    As a side note I have to mention that whenever a player with a pencil and a note pad showed up on a craps table one of the old time box men(Vegas term for the old timers)  would say, "Here we go, another jagoff off from providence".  "Wait fro 35 rolls without a 12  amd then load up on it."  That was the craps knowledge of many from Providence.  The saying about players from Providence came about because many of those old timers had ran the games there.  Duh. 

    providence, along with Saratoga, French Lick, and many other places had illegal gambling and some geniuses from Providence that convinced other players to show up with pencils and notebooks didn't even know that tables that were labeled "Don't Pass Bar  3* weren't legit games. Nothing do with all illegal games but those who ran legit ones wanted their players to come back. 

    *The 3 on the layout was shown as a die showing one pip and a die showing two.

    Vegas gets which off of players who think they know how to play. 

    Thye lotteries don't really need any help but those who honestly think they can reduce the odds are their favorite players."
• db101

  

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  Jan 31, 2023, 11:13 pm

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  In response to eddessaknight

  Quote: Originally posted by eddessaknight on Jan 31, 2023

  This informative  Dice odds contribution credit  by,  our professional gaming LV member

  COIN TOSS:

  • "For a real good lesson in odds consider how the odds of a pair of dice (each die with six pips on it)  rollings 2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12  come into being. 

    There's one and only one way to roll a 2, a one and a one. Two ways to roll a 3, a one and a two or a two and a one. Go from there.  I'll help you out, 4, 5, ,and 6 are the same as 8,l9,and 10 for odds purposes.Six ways to roll a 7. 

    As a side note I have to mention that whenever a player with a pencil and a note pad showed up on a craps table one of the old time box men(Vegas term for the old timers)  would say, "Here we go, another jagoff off from providence".  "Wait fro 35 rolls without a 12  amd then load up on it."  That was the craps knowledge of many from Providence.  The saying about players from Providence came about because many of those old timers had ran the games there.  Duh. 

    providence, along with Saratoga, French Lick, and many other places had illegal gambling and some geniuses from Providence that convinced other players to show up with pencils and notebooks didn't even know that tables that were labeled "Don't Pass Bar  3* weren't legit games. Nothing do with all illegal games but those who ran legit ones wanted their players to come back. 

    *The 3 on the layout was shown as a die showing one pip and a die showing two.

    Vegas gets which off of players who think they know how to play. 

    Thye lotteries don't really need any help but those who honestly think they can reduce the odds are their favorite players."

  That's all completely irrelevant to this puzzle because we're not talking about your standard numbered dice here. But speaking of regular dice, I got another really good probability puzzle courtesy of my whiz kid. I'll try to remember to post it tomorrow.

  PS- Old Sausage Fingers is the last person you should be quoting in regard to anything mathematical. A person can know a lot about games and gambling without knowing a lick of math. That dude's in the remedial class when it comes to math. See exhibit A https://www.lotterypost.com/thread/344172/7169123