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Mental Aerobics…

Posted By: Randy GageFebruary 13, 2009

Since it’s Friday and a lot of you will soon be vegging out for the weekend, let’s give your brain a little exercise first.  Here’s a little challenge for you:

David and Brian are serving a prison sentence for chewing gum in public in Singapore.  They share a cell and spend their days rolling dice to pass the time. They each have one die.  They both are so worn out however, that only three sides are visible.   David’s die shows only the sides 2, 4 and 5.  Brian’s die shows only 1, 3 and 6.  If their game awards the winner who rolls the highest number each time, who will win most often over the long run?

Post your solution below.  The first one to get the correct answer receives the worldwide fame associated with posting a comment on my blog!

-RG

44 comments on “Mental Aerobics…”

  1. David will win. 3 of his numbers beat 2 of Brian's. Brian has to roll a six to win. David has 3 chances to beat Brian on every roll but a six and is guaranteed to win on every roll of a 1 by Brian.

  2. that would be David - his average through will be - (2+4+5)/6 = 11/6. Brian's average through will be - (1+3+6)/6 = 10/6.

  3. David would have the highest average because of his dice having the 2&4, but knowing you, Randy, there has to be a trick answer somewhere. Averaging out the number of rolls would probably end up with neither of them having the most consistency in the wins though. So it would most likely be neither.

  4. David wins.

    2, 4 and 5 vs 1, 3 and 6? Let's see:

    David: 2 wins against 1. 4 wins against 1 or 3. 5 also wins against 1 and 3. Five wins per "statistically probable cycle".

    Brian: 3 wins against 2. 6 wins against 2, 4 and 5. Four wins. Only four wins per "statistically probable cycle".

    I have no idea if that's a mathematically valid term, or if it makes sanse at all - but that's my reasoning.

    Have a great weekend and valentine's, folks!

    /Niclas

  5. Okay all, I was trying to use a little levity, since Singapore regulates everything and is kind of like Disney world, but with the death penalty. But that's not a trick to fool you. The challenge is figuring out who will win the dice toss.

    -RG

  6. I did a quick evaluation and choose David. Call it intuition or common sense or whatever, but I didn't do numbers and things to back me up. 🙂 This reminds me of LSAT. 🙂

  7. Mhh... definitely David. Though I'm still wondering if this is a trick question... then I'll have to agree with Carole... No one can be arrested for chewing gum in a public place in Singapore.

  8. If they are actually each only rolling one die, then even if the paint is worn off you can still count the drill holes...or know which side came up by seeing what is on the opposite side of the blank.
    Meaning each one has the same chance to win since technically they are still each rolling a die with 6 sides.

    Or...you said they are rolling "dice"...so they will still each roll both dice on their turn. Still meaning there is not a way statistically to know who will win.
    Jeanne

  9. Randy wins for getting all of us to read this and post our responses and drive up his SEO and I am fine with that lesson.

    Would love for everyone to join my Google friends network.

    Steve

  10. Isn't the law in Singapore for gum related offenses imprisonment for spitting your gum on the sidewalk? I mean if it's just for chewing, how smart do you have to be to swallow it when you see the cops coming after you during a chew?

    I know, I know. That wasn't the question. But you have to admit, it does raise an interesting point about David & Brian's intelligence level.

  11. I'll use a few overused cliches to answer the question.

    You can make up in numbers what you lack in skill;

    Don't worry about the cards (Dice) you were dealt, just be grateful you have any at all;

    Success does not depend on what you have, it depends on what you do; (I don't know where that came from but it sounds pretty good..lol)

    In other words it doesn't matter who has the highest numbers showing. Whoever rolls the dice more times than the other person will win. Don't pay attention to the numbers on the dice, or the things you can't control. Just work harder and roll the dice more often than anyone else and you win whatever game you're playing.

  12. This is an addendum to my answer.

    I assume you mean that Brian and David are both worn out and all 6 sides are on their dice. They both have the same die because no matter how you look at a die, you can only see 3 sides unless you have a mirror.

    My answer is still the same. Neither one of them should worry about what they can see because they should know that they have the same die even though they can't see it.

    What you can see is not important. What you know is more important.

    So, just like I said before. Whoever rolls more often will win.

  13. This are David's chances=(1/2)*(0)+(1/6)*2+(1/6*4)+(1/6)*5 = 1.833
    These are Brian's chances =(1/2)*(0)+(1/6)*1+(1/6*3)+(1/6)*6 = 1.66

    Half of the times both get Zero and if the other gets any numbered side they will win, the balance moves towards David when they both nail numbered sides.

    I guess I am not even close to getting the chance to post since a lot of people already have David!

    Keep em coming!

    Federico

  14. Actually I think statistics show that 7 is usually the most common number rolled and next to follow is 6 and 8. Since no one has a 7 or an 8 then 6 would be the most common roll.
    Therefore, Brian would win.

  15. David is the winner over Brian because David will have the advantage of 5 higher numbers versus Brian's 4 in average,
    in the long run. The 6's cannot win it for Brian as they are just
    one of 4.

  16. Well, if they roll the dice at the same time, and the same amount of times, I have to say that David will win, but if it is not like that, then the one who rolls the dice the most is the winner.

  17. Logically, it should be David. Because two numbers are higher than Brian. 2 out of 3 rolls (when numbers are shown on both sides are highter).
    If we go by chance, either David of Brian could win, dependind what number come out.
    Brian could roll a 1 all day, and David could roll a worn side all day. Still making Brian a winner.
    If this is not the answer, I am anxious to know.
    Carole

  18. Randy,
    I just want to point out that its not legal to sell gum in Singapore. However you can buy it overseas and chew it in Singapore.

  19. David or Brian
    They are both worn out, but whoever rolls more dice will end up wining. Persistence is the key. Don't stop until you won. Who will will in the long run? David or Brian. If i was david, it would be David. If i was Brian, it would be Brian.

    Butch Gagnon

  20. david's dice will show 3 number:

    if it is 2:brian's dice will show(1,3 or 6)so he has 2/3 chance to win.

    if it is 4:brian's dice will show(1,3 or 6)so he has 1/3 chance to win.

    if 5:again brian has 1/3 chance.

    totally brian has 4/3 chance.

    for brian:

    if he got:

    1:david's chance 3/3

    3:david chance 2/3

    6:david's chance is zero.totally david has 5/3 chance.

    so david will be winner.

  21. Dear Randy

    I love you,you changed my mind,life and everything in my life.
    these sentences of you always is the front of my eyes.

    "live life,live your life and make it a life worth living"

    thanks for everything.

  22. Well, they each have four numbers really, because the opposite sides always add to seven. So if David rolls a a blank and the side resting on the bottom is a four you know he rolled a three. Same idea for Brian's die except with a three.

    However, this has nothing to do with the solution. The solution is that neither has a higher probability of winning. People are using deductive reasoning by stating one die has a number that beats three of the others.

    Inductive reasoning, the reasoning you should be working off of, would tell you that the numbers don't matter so much as them actually rolling the die. Each die has the same probability of rolling on each number. So in the end, the winner is chosen by chance, not by what number is shown.

    Merry Christmas *Waves*

  23. acctually the winner is whomever rolls the highest number each time. each time they roll the dies, they each have the abbility to outroll their opponent, the same as if the dies were full.
    we tend to try to dig too deep, and make it more complicated for ourselves, even though the answer is there. "who will win each time after they roll?" The highest number wins lol

  24. I assume that if either die shows up as blank, it's a tie. therefor ethere are 9 possible outcomes to a toss:

    Brian David winner
    1 2 d
    1 4 d
    1 5 d
    3 2 b
    3 4 d
    3 5 d
    6 2 b
    6 4 b
    6 5 b

    as you can see d (David) wins 5 of 9 times. Since each toss is equally likely to occur, David will win more in the long run.

    Randy I claim the fame!

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  • 44 comments on “Mental Aerobics…”

    1. David will win. 3 of his numbers beat 2 of Brian's. Brian has to roll a six to win. David has 3 chances to beat Brian on every roll but a six and is guaranteed to win on every roll of a 1 by Brian.

    2. that would be David - his average through will be - (2+4+5)/6 = 11/6. Brian's average through will be - (1+3+6)/6 = 10/6.

    3. David would have the highest average because of his dice having the 2&4, but knowing you, Randy, there has to be a trick answer somewhere. Averaging out the number of rolls would probably end up with neither of them having the most consistency in the wins though. So it would most likely be neither.

    4. David wins.

      2, 4 and 5 vs 1, 3 and 6? Let's see:

      David: 2 wins against 1. 4 wins against 1 or 3. 5 also wins against 1 and 3. Five wins per "statistically probable cycle".

      Brian: 3 wins against 2. 6 wins against 2, 4 and 5. Four wins. Only four wins per "statistically probable cycle".

      I have no idea if that's a mathematically valid term, or if it makes sanse at all - but that's my reasoning.

      Have a great weekend and valentine's, folks!

      /Niclas

    5. Okay all, I was trying to use a little levity, since Singapore regulates everything and is kind of like Disney world, but with the death penalty. But that's not a trick to fool you. The challenge is figuring out who will win the dice toss.

      -RG

    6. I did a quick evaluation and choose David. Call it intuition or common sense or whatever, but I didn't do numbers and things to back me up. 🙂 This reminds me of LSAT. 🙂

    7. Mhh... definitely David. Though I'm still wondering if this is a trick question... then I'll have to agree with Carole... No one can be arrested for chewing gum in a public place in Singapore.

    8. If they are actually each only rolling one die, then even if the paint is worn off you can still count the drill holes...or know which side came up by seeing what is on the opposite side of the blank.
      Meaning each one has the same chance to win since technically they are still each rolling a die with 6 sides.

      Or...you said they are rolling "dice"...so they will still each roll both dice on their turn. Still meaning there is not a way statistically to know who will win.
      Jeanne

    9. Randy wins for getting all of us to read this and post our responses and drive up his SEO and I am fine with that lesson.

      Would love for everyone to join my Google friends network.

      Steve

    10. Isn't the law in Singapore for gum related offenses imprisonment for spitting your gum on the sidewalk? I mean if it's just for chewing, how smart do you have to be to swallow it when you see the cops coming after you during a chew?

      I know, I know. That wasn't the question. But you have to admit, it does raise an interesting point about David & Brian's intelligence level.

    11. I'll use a few overused cliches to answer the question.

      You can make up in numbers what you lack in skill;

      Don't worry about the cards (Dice) you were dealt, just be grateful you have any at all;

      Success does not depend on what you have, it depends on what you do; (I don't know where that came from but it sounds pretty good..lol)

      In other words it doesn't matter who has the highest numbers showing. Whoever rolls the dice more times than the other person will win. Don't pay attention to the numbers on the dice, or the things you can't control. Just work harder and roll the dice more often than anyone else and you win whatever game you're playing.

    12. This is an addendum to my answer.

      I assume you mean that Brian and David are both worn out and all 6 sides are on their dice. They both have the same die because no matter how you look at a die, you can only see 3 sides unless you have a mirror.

      My answer is still the same. Neither one of them should worry about what they can see because they should know that they have the same die even though they can't see it.

      What you can see is not important. What you know is more important.

      So, just like I said before. Whoever rolls more often will win.

    13. This are David's chances=(1/2)*(0)+(1/6)*2+(1/6*4)+(1/6)*5 = 1.833
      These are Brian's chances =(1/2)*(0)+(1/6)*1+(1/6*3)+(1/6)*6 = 1.66

      Half of the times both get Zero and if the other gets any numbered side they will win, the balance moves towards David when they both nail numbered sides.

      I guess I am not even close to getting the chance to post since a lot of people already have David!

      Keep em coming!

      Federico

    14. Actually I think statistics show that 7 is usually the most common number rolled and next to follow is 6 and 8. Since no one has a 7 or an 8 then 6 would be the most common roll.
      Therefore, Brian would win.

    15. David is the winner over Brian because David will have the advantage of 5 higher numbers versus Brian's 4 in average,
      in the long run. The 6's cannot win it for Brian as they are just
      one of 4.

    16. Well, if they roll the dice at the same time, and the same amount of times, I have to say that David will win, but if it is not like that, then the one who rolls the dice the most is the winner.

    17. Logically, it should be David. Because two numbers are higher than Brian. 2 out of 3 rolls (when numbers are shown on both sides are highter).
      If we go by chance, either David of Brian could win, dependind what number come out.
      Brian could roll a 1 all day, and David could roll a worn side all day. Still making Brian a winner.
      If this is not the answer, I am anxious to know.
      Carole

    18. Randy,
      I just want to point out that its not legal to sell gum in Singapore. However you can buy it overseas and chew it in Singapore.

    19. David or Brian
      They are both worn out, but whoever rolls more dice will end up wining. Persistence is the key. Don't stop until you won. Who will will in the long run? David or Brian. If i was david, it would be David. If i was Brian, it would be Brian.

      Butch Gagnon

    20. david's dice will show 3 number:

      if it is 2:brian's dice will show(1,3 or 6)so he has 2/3 chance to win.

      if it is 4:brian's dice will show(1,3 or 6)so he has 1/3 chance to win.

      if 5:again brian has 1/3 chance.

      totally brian has 4/3 chance.

      for brian:

      if he got:

      1:david's chance 3/3

      3:david chance 2/3

      6:david's chance is zero.totally david has 5/3 chance.

      so david will be winner.

    21. Dear Randy

      I love you,you changed my mind,life and everything in my life.
      these sentences of you always is the front of my eyes.

      "live life,live your life and make it a life worth living"

      thanks for everything.

    22. Well, they each have four numbers really, because the opposite sides always add to seven. So if David rolls a a blank and the side resting on the bottom is a four you know he rolled a three. Same idea for Brian's die except with a three.

      However, this has nothing to do with the solution. The solution is that neither has a higher probability of winning. People are using deductive reasoning by stating one die has a number that beats three of the others.

      Inductive reasoning, the reasoning you should be working off of, would tell you that the numbers don't matter so much as them actually rolling the die. Each die has the same probability of rolling on each number. So in the end, the winner is chosen by chance, not by what number is shown.

      Merry Christmas *Waves*

    23. acctually the winner is whomever rolls the highest number each time. each time they roll the dies, they each have the abbility to outroll their opponent, the same as if the dies were full.
      we tend to try to dig too deep, and make it more complicated for ourselves, even though the answer is there. "who will win each time after they roll?" The highest number wins lol

    24. I assume that if either die shows up as blank, it's a tie. therefor ethere are 9 possible outcomes to a toss:

      Brian David winner
      1 2 d
      1 4 d
      1 5 d
      3 2 b
      3 4 d
      3 5 d
      6 2 b
      6 4 b
      6 5 b

      as you can see d (David) wins 5 of 9 times. Since each toss is equally likely to occur, David will win more in the long run.

      Randy I claim the fame!

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