1. ## 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?

2. ## Re: Match Up

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

3. ## 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>

4. ## 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.

5. ## Re: Match Up

<span style="background-color: #FFFF00; color: #FFFF00; font-weight: bold">15.625%</span hide>

6. ## 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?

7. ## 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>

8. ## 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?

9. ## 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. ## 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>

11. ## Re: Match Up

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

12. ## 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>

13. ## Re: Match Up

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

14. ## 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

15. ## 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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