Odds of Manchester United vs Aston Villa in FA Cup 3rd Round

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Discussion Overview

The discussion revolves around calculating the odds of Manchester United being drawn against Aston Villa in the 3rd round of the FA Cup, specifically focusing on the frequency of this occurrence in past draws. The scope includes mathematical reasoning and probability calculations related to sports draws.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant states that Manchester United has been drawn against Aston Villa 4 times in the past 7 draws and seeks to verify the odds of this happening.
  • Another participant proposes an answer of 1/472,216 and asks for verification.
  • A different participant suggests that to ensure understanding, one should post the correct method for calculating the odds.
  • One participant calculates the probability of Manchester United drawing Aston Villa 4 times followed by 3 failures, arriving at a probability of 1/472,216 based on permutations.
  • Another participant presents an alternative calculation resulting in 1191640/562711519881, noting that this is close to the previous answer and provides additional probabilities for drawing any team 4 times out of 7.
  • This participant also mentions the likelihood of at least one team being drawn against another team 4 times, providing further context to the calculations.

Areas of Agreement / Disagreement

There is no consensus on the correct probability calculation, as participants present differing methods and results. Multiple competing views remain regarding the odds of the draws.

Contextual Notes

The calculations depend on the assumptions made about the independence of draws and the specific teams involved. There are unresolved mathematical steps and differing interpretations of the probability outcomes.

billiards
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A friend recently observed that Manchester United have been drawn against Aston Villa in The 3rd round of the FA Cup 4 times in the past 7 draws. There are 64 teams in the 3rd round draw. What are the odds of this happening?

I believe I have the answer, but I need someone to verify my answer to ensure that my understanding is correct.

So, what are the odds?
 
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Okay, obviously this is too easy a question to bother most of you. My answer was 1/472,216. Is that right?
 
Actually, I think the problem is here

I believe I have the answer, but I need someone to verify my answer to ensure that my understanding is correct.

If you want to make sure your understanding is correct, the best way is surely to post what you think the correct method would be
 
Okay.

Chance of Man U drawing Villa on one occasion A = 1/63
Chance of that not happening on same occasion B = 62/63

Chance of it happening 4 times in a row followed by 3 failures is AAAABBB=623/637.

This probability is equal to the probability of the initial condition being satisfied (that Manu draw Vill 4 times in 7), by each of the 7!/(4!*3!) = 35 different possible permeatations. Accounting for all possible solutions, the probability that the initial condition is satisfied, is 35 times 623/637 = 1/472,216.
 
Looks good to me
 
I get 35*62^3/63^7 = 1191640/562711519881, but that's close enough to your answer.

Note that the chance of Man U drawing some team 4 times out of 7 (not just Villa in particular) is 1191640/8931928887, which at roughly 1/7495 is a fair bit more likely. The chance of it happening to at least one team (not just Man U) is about 1/242. The chance of at least one team pulling some other team at least 4 times is little changed at about 1/238.
 

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