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Mental Aerobics Solution

Posted By: Randy GageFebruary 14, 2009

Quite a response on the mental aerobics challenge I gave you.  Should have known this would bring Richard Avon out of the woodwork!  Even my nephew Zach.    So your challenge was:

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?

The Solution:

Answer:  David will win 55% of the time.  Here’s why:

When David rolls 2, he will win once and lose twice.
When David rolls 4, he will win twice, and lose once.
When David rolls 5, he will win twice and lose once.
When Brian rolls 1, he will lose three times
When Brian rolls 3, he will win once, and lose twice.
When Brain rolls 6, he will win three times.
So David wins 5 times, and Brian only four.

Quite a few of you got it right, although the logic of some of you was a little shaky.  And a few of you have conspiracy theory issues!  The fame and adulation for being first goes to Michael Eisbrener.

-RG

11 comments on “Mental Aerobics Solution”

  1. HI Randy,

    I feel compelled to comment here...

    First, assuming your solution is correct, then 5 out of 9 is 55.55555... %. If you are rounding this to the nearest full number then it is 56%.

    However, neither the problem nor the solution makes any comment on what happens when the die is rolled to a blank surface. Your solution is only correct if the rules allow for that die to be rolled again - however, the problem does not suggest that.

    When David rolls 2 then he wins four times and loses twice
    When David rolls 4, he will win five times, and lose once.
    When David rolls 5, he will win five times and lose once.
    When Brian rolls 1, he will lose three times and wins three times
    When Brian rolls 3, he will win four times, and lose twice.
    When Brain rolls 6, he will win six times.
    So David wins 14 times, and Brian only 13.
    yielding a result of 51.85% in favour of David.

    Of course, that is an extremely simple approach, which gives only an approximate answer, and is not mathematically / statistically accurate.

    So - as the problem is written, we have to allow for the blank sides to count as zero. If this is the case, then the ONLY truly correct answer provided was that of Federico. The end result is that David wins 52.52% of the time.

    Hate to be pedantic - actually, - no I don't... 😉

    My apologies to Michael, but your answer is not correct, as you go in to detail which does not allow for the blank sides.

    Because David did not go into any incorrect detail, his is actually the first (approximately) correct answer...

    Regards,
    Trevor

  2. Ok, I'm definitely Monday morning quarterbacking here, but when I first saw the question, I just figured that David's average was higher than Brians', so of course he would win over time.

    But then I figured that the zeros were some trick part of the question, like "You can't know unless you know all the rules", and I didn't bother.

    But thanks for the game!

  3. P.S

    My life long friend Bill Wynn whom I met on your forum STILL ste each other this sortt of thing now and again and last week I sent him this to which he hasn't YET replied!

    What would you do with this

    hijklmno?

  4. Here is the problem -

    Rolling a 1 will win three times out of six, because the other die has three zeros.

    Trevor

  5. Randy -

    I would have thought you would get that one...

    "h" to "o" progressively through the alphabet...

    Well done Richard...

    Regards,
    Trevor

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  • 11 comments on “Mental Aerobics Solution”

    1. HI Randy,

      I feel compelled to comment here...

      First, assuming your solution is correct, then 5 out of 9 is 55.55555... %. If you are rounding this to the nearest full number then it is 56%.

      However, neither the problem nor the solution makes any comment on what happens when the die is rolled to a blank surface. Your solution is only correct if the rules allow for that die to be rolled again - however, the problem does not suggest that.

      When David rolls 2 then he wins four times and loses twice
      When David rolls 4, he will win five times, and lose once.
      When David rolls 5, he will win five times and lose once.
      When Brian rolls 1, he will lose three times and wins three times
      When Brian rolls 3, he will win four times, and lose twice.
      When Brain rolls 6, he will win six times.
      So David wins 14 times, and Brian only 13.
      yielding a result of 51.85% in favour of David.

      Of course, that is an extremely simple approach, which gives only an approximate answer, and is not mathematically / statistically accurate.

      So - as the problem is written, we have to allow for the blank sides to count as zero. If this is the case, then the ONLY truly correct answer provided was that of Federico. The end result is that David wins 52.52% of the time.

      Hate to be pedantic - actually, - no I don't... 😉

      My apologies to Michael, but your answer is not correct, as you go in to detail which does not allow for the blank sides.

      Because David did not go into any incorrect detail, his is actually the first (approximately) correct answer...

      Regards,
      Trevor

    2. Ok, I'm definitely Monday morning quarterbacking here, but when I first saw the question, I just figured that David's average was higher than Brians', so of course he would win over time.

      But then I figured that the zeros were some trick part of the question, like "You can't know unless you know all the rules", and I didn't bother.

      But thanks for the game!

    3. P.S

      My life long friend Bill Wynn whom I met on your forum STILL ste each other this sortt of thing now and again and last week I sent him this to which he hasn't YET replied!

      What would you do with this

      hijklmno?

    4. Here is the problem -

      Rolling a 1 will win three times out of six, because the other die has three zeros.

      Trevor

    5. Randy -

      I would have thought you would get that one...

      "h" to "o" progressively through the alphabet...

      Well done Richard...

      Regards,
      Trevor

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