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Thread: Match Up

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    5 Star Lounger Ruff_Hi's Avatar
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    Match Up

    64 people enter a knockout tournament. The first round consists of 32 matches between 2 competitors. The 32 winners progress to the second round. The second round consists of 16 matches between 2 competitors - the 16 winners progress to the third round. Etc Etc until only 1 competitor is undefeated.

    If the 64 competitors are of roughtly equal ability and the opponents in each round are randomly selected, what is the probability that competitor A will compete against competitor B during the tournament?
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

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    Re: Match Up

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1 in 32 (or 3.125 %)</span hide>

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    5 Star Lounger Ruff_Hi's Avatar
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    Re: Match Up

    <img src=/S/hmmn.gif border=0 alt=hmmn width=15 height=15>That is not the answer that I have. Isn't that the <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">answer that they will meet in the first round? What about the possibility that they meet in the second, third, etc rounds?</span hide>
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

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    Re: Match Up

    I did take the 2nd, 3rd etc. rounds into account, but apparently my calculation is different from yours. Let's see what others make of it.

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    Re: Match Up

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">15.625%</span hide>
    <img src=/w3timages/blueline.gif width=33% height=2>
    <big>John</big>

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    5 Star Lounger Ruff_Hi's Avatar
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    Re: Match Up

    Ok, I've reviewed my solution and I think I have found a hole. The new way I have worked out if they meet in the first round is ...

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Divide the 64 into two equal groups (G1 and G2) of 32. Place one person from G1 to matches 1 to 32 so that each match has one competitor. Randomly allocate one person from G2 to each match. Now, the four equally likely outcomes from the division into two groups is ...
    1. <LI>A in G1, B in G1
      <LI>A in G1, B in G2
      <LI>A in G2, B in G1
      <LI>A in G2, B in G2
    The probability that A will meet B in round 1 under 1) and 4) above is zero. The probability that A will meet B in round 1 under 2) and 3) above is 1/32. Thus the total probability that A will meet B in round 1 is 1/2 x 0 + 1/2 x 1/32 = 1/64</span hide>

    Hans - is that the answer you got for meeting in round 1?
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

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    Re: Match Up

    My calculation for the 1st round was <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">A has 64-1 = 63 potential opponents in the first round, so the probability of playing against B is 1/63.</span hide>

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    5 Star Lounger Ruff_Hi's Avatar
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    Re: Match Up

    yup - I've just run a simulation (500,000 trials) and it supports <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/63</span hide> more than <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/64</span hide> - wonder where the new hole in my thinking is?
    (Location Australia, then UK, but now USA. Heart, outlook, attitude, etc always Australian)
    Quote: "All Happiness is the release of internal pressure"

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    Re: Match Up

    I get the same answer as Hans:
    <table border=1><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Round</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Prob</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Calc</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1.587%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=1/63</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">2</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.794%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=62/63/4/31</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">3</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.397%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=62/63/4*30/31/4/15</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">4</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.198%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=62/63/4*30/31/4*14/15/4/7</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">5</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.099%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=62/63/4*30/31/4*14/15/4*6/7/4/3</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">6</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">0.050%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">=62/63/4*30/31/4*14/15/4*6/7/4*2/3/4</span hide></td><tr><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Total</span hide></td><td align=center valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">3.125%</span hide></td><td valign=bottom><span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">

  10. #10
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    Re: Match Up

    Not only the same answer, exactly the same calculation too! <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

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    Re: Match Up

    Do you need to take into account the apparent(?) 50% chance that either A or B will get knocked out?

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    Re: Match Up

    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">Yep, that's the /4 that occurs repeatedly: the chance that both A and B will make it to the next round is 1/2 * 1/2 = 1/4</span hide>

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    Plutonium Lounger Leif's Avatar
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    Re: Match Up

    Ah! - that answers my next question then <img src=/S/grin.gif border=0 alt=grin width=15 height=15>

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    Re: Match Up

    I worked this out a bit differently, but it seems to give the same answer...

    In each round the chance of both remaining in the tournament is
    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/1 (a certainty in round 1), 1/4, 1/16, 1/64, 1/256, 1/1024 (for round 6)</span hide>
    In each round the chance of them meeting, if they are still both in is
    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">1/63, 1/31, 1/15, 1/7, 1/3, 1/1 (a certainty in round 6)</span hide>
    So the total ods of them meeting is
    <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">(1 * 1/63) + (1/4 * 1/31) + (1/16 * 1/15) + (1/64 * 1/7) + (1/256 * 1/3) + (1/1024 * 1)</span hide>
    which gives a result of <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">3.2615%</span hide>

    StuartR

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    Re: Match Up

    <hr>but it seems to give the same answer...<hr>

    <img src=/S/confused.gif border=0 alt=confused width=15 height=20> When has <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">3.2615%</span hide> been equal to <span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold"> 3.125%</span hide>?

    Steve

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