Probability that the Patriots Cheated!

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bquinn

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For anyone living under a rock this past week, the "deflategate " argument has been front and center in the sports media, but I have been unable to find a statistic that shows probability.

If you are unaware of the circumstance, in last weeks AFC championship game, it was found that 11 out of 12 of the Patriots footballs were under inflated. Their opponents, the Colts had zero footballs under inflated. There is an advantage to playing with an under inflated ball (I can explain if you care to hear the explanation). Recently, there has been some "scientific" explanations reasoning that the atmospheric pressure and temperature changes can alter the PSI of the ball. While I understand this to be true, I'd like to see the formula/explanation that shows the probability that the 11 under inflated balls out of 24 tested all belong to the Patriots. Thanks!
 
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bquinn said:
For anyone living under a rock this past week, the "deflategate " argument has been front and center in the sports media, but I have been unable to find a statistic that shows probability.

If you are unaware of the circumstance, in last weeks AFC championship game, it was found that 11 out of 12 of the Patriots footballs were under inflated. Their opponents, the Colts had zero footballs under inflated. There is an advantage to playing with an under inflated ball (I can explain if you care to hear the explanation). Recently, there has been some "scientific" explanations reasoning that the atmospheric pressure and temperature changes can change the PSI of the ball. While I understand this to be true, I'd like to see the formula/explanation that shows the probability that the 11 under inflated balls out of 24 tested all belonged to the Patriots. Thanks!
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One would need to know what the probability that a given football is under inflated is. I would think it is a low number but how low?
 
Jomo said:
One would need to know what the probability that a given football is under inflated is. I would think it is a low number but how low?
Click to expand...

well I'm not entirely sure that's necessary. Given the fact that 11 out of the 24 balls tested happened to be under inflated, I'm curious as to how likely in this event that all 11 of the under inflated balls belonged to the patriots. With further thinking, I'm using coin flip probability to calculate all of the possible outcomes of dividing 24 balls between 2 teams. So...

1
2^24

=

1
16,777,216

With my math, that means there's 16,777,216 possible outcomes. However, there are 12 total outcomes that allow the Patriots to have 11 under inflated balls and 1 regularly inflated balls, which means...

12
16,777,216 = .0000007152 = .00007153% chance that all 11 of the 24 under inflated balls belong to them! Not looking so good!


However, I'm pretty far removed from statistics/probability calculations and could have forgotten something. So please correct me if I'm wrong. THANKS!!
 
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bquinn said:
well I'm not entirely sure that's necessary. Given the fact that 11 out of the 24 balls tested happened to be under inflated, I'm curious as to how likely in this event that all 11 of the under inflated balls belonged to the patriots. With further thinking, I'm using coin flip probability to calculate all of the possible outcomes of dividing 24 balls between 2 teams. So...

1
2^24

=

1
16,777,216

With my math, that means there's 16,777,216 possible outcomes. However, there are 12 total outcomes that allow the Patriots to have 11 under inflated balls and 1 regularly inflated balls, which means...

12
16,777,216 = .0000007152 = .00007153% chance that all 11 of the 24 under inflated balls belong to them! Not looking so good!


However, I'm pretty far removed from statistics/probability classes and VERY EASILY could have forgotten something. So please correct me if I'm wrong. THANKS!!
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If you flipped a coin 24 times then there are 2^24 = 16777216 outcomes as you stated.

Let H be inflated properly and T be under inflated.

So you want under inflated to come up 11 times in the 1st 12 times and then correctly inflated 12 more times. So a typical sequence would be (using H and T as defined above) THTTTTTTTTTTHHHHHHHHHHHH. Since you can put the 1st H in any one of 12 positions there are 12 possible outcomes, just as you stated. If the probability of being under inflated is p then P(THTTTTTTTTTTHHHHHHHHHHHH)=[p^11(1-p)^13]/16777216. Since there are 12 outcomes with the 1st 12 having 11 under inflated followed by 12 correctly inflated football I would think that the probability would be 12[p^11(1-p)^13]/16777216. I am fairly certain that this is correct but you never know.

I do want to point out that the exact value of the prob that the ball is under inflated matters. If a randomly selected football had a 99% chance of being under inflated then I would be suspicious of the team that had 12 correctly inflated balls (maybe they thought incorrectly that a properly inflated ball would be to their advantage) and NOT the Patriots!!

EDIT: Please assume independence for the balls being under inflated
 
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Thanks for clarifying the formulas! I do realize that the probability a ball loses pressure due to environmental conditions should be way less than 11/24 (46%) and further hurt the patriots odds. However, I'm specifically trying to refute their position that it happened naturally, which in this case all 24 balls were in the same environmental conditions, and 11 lost air. With those factors being debated as a legitimate possibility, I was curious as to what the chances are all of the compromised balls belong to the patriots. I'd say it's safe to say the probability of chance is a long shot! .00007153% to be exact. Cheaters!
 
Jomo said:
If you flipped a coin 24 times then there are 2^24 = 16777216 outcomes as you stated.

Let H be inflated properly and T be under inflated.

So you want under inflated to come up 11 times in the 1st 12 times and then correctly inflated 12 more times. So a typical sequence would be (using H and T as defined above) THTTTTTTTTTTHHHHHHHHHHHH. Since you can put the 1st H in any one of 12 positions there are 12 possible outcomes, just as you stated.


So the way that I'm looking at this problem, 2^24 represents the number of different Patriot/Colt combinations available for the 24 available footballs, 11 of which are under-inflated. Lets say the first 11 footballs are under-inflated, and the last 13 are properly inflated. T represents under-inflated, H properly inflated:

TTTTTTTTTTTHHHHHHHHHHHHH

So what I'm saying is that for each T or H above, there are two possibilities: Pats (P) or Colts (C). Hence the 2^24. We want the Pats to have all 11 under-inflated balls plus one of the properly inflated balls. A sample combination would be:


P P P P P P P P P P P P C C C C C C C C C C C C

Another possibility would be:

P P P P P P P P P P P C C C C C P C C C C C C C

My point is that there should be 13 different ways the Pats can have 11 under-flated footballs and 1 properly inflated, not 12 ways (because the Pats can have any one of the 13 properly inflated footballs). Let me know what you think, this is just the way the way I'm seeing it. Could be wrong! Thanks!
Click to expand...
 
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