Uh oh, here we go: CBS claims Pats win coin flip at "impossible clip"

PlainOldEd said:

what is 13 out of 25 run through the same formula?

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The probability of getting exactly 13 out of 25 tosses correct is 15.5%. The probability of getting 13+ tosses correct is - as expected - exactly 50%.

PlainOldEd said:

I am no mathematician, but this doesn't seem right to me. First off, wouldn't it be 1 in 730? but even that seems wrong. If you flipped a coin multiple times you would expect that over time you would win about 1/2 but if you calculate the odds of flipping a coin 25 times and winning 12 or 13 I believe it comes out to around 15%. Again, I'm not a math guy but I suspect this is a case of fuzzy math....
You know the old saying that goes something like this... ?

There are Lies, Damn Lies and Statistics.

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1/0.0073≈137

1 in 137. Improbable, but not remarkable. If you have thousands of people scrutinizing the patriots trying to find statistical anomalies, you're probably going to find a few dozen of these curiously improbable statistics just by chance.

Rob0729 said:

There are plenty of variables to consider:

• Did the Pats always call heads or tails? If they didn't call the same side of the coin each time then it isn't 50-50.
• Did the opposing teams in games at Gillette (the away team calls the coin flip) always call the side of the coin opposite of what the Pats always call? Say if the Pats always call heads on the road and the away team called heads and it landed on tails, the Pats winning the coin flip was actually statistically a loss since probability is 50-50 that it will always land on heads or tails not whether the Pats win the coin toss if they don't have exclusively the same side of the coin.
• Did each ref flip the coin the same way? If one refs flips the coin and the coin spins a lot in the air while another ref flips the coin and it doesn't spin at all, that will affect the probabilities of how the coin lands. The 50-50 probability is only when the coin is flipped exactly the same every time.
• Were any of the flips in extreme wind or rain? Extreme weather can affect a coin flip.

Those are just some examples of variables affecting the outcome. The 50-50 thing is not a constant. If you flip a coin 10 times, odds are good you will not get 5 heads and 5 tails. Variables will always affect the probability unless you can recreate each flip perfectly down to the detail.

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Points 1 and 2 wouldn't affect the probability.

Points 3 and 4 could.

Based on research, the probability would be 51% biased towards whichever side was up when the coin was thrown into the air. Also, if the coin spins in some way, there's as high as an 80% chance that it lands on on the heavier side.

Scenario 2 is unlikely in this case, and in scenario 1 the probability isn't that far off from 50%.

The formula that leads to the probability of 0.0073 is based on a fair coin. In reality, the coin may not be 100% fair due to imperfections, but that's still a pretty close approximation.

The fact that the Pats don't always get to choose is all we need to show that there is no cheating.

Some basic probability:

The odds of this happening are 1 in 136.
There are 32 teams. That means there is a 1 in 4.25 chance it happens to any one team.
That is, there is about a 25% chance this happens in the league every 25 games.
There's about a 66% chance it happens once every six seasons.
It is not impossible. It is not unlikely. It is probable. It is more probable than not that this happens every six seasons.

Now, clearly, someone pulled a specific sample set - 19 of the last 25. Had the Patriots won the toss 26 games ago, they would have said 20 of the last 26. So we can surmise that the odds are slightly lesser, at 19 of the past 26. The pattern prior to the toss 26 games ago might appear more random. Although I guess one could argue that 25 games ago the Patriots learned to cheat the coin toss.

Interestingly, fraud detection counts on the human presumption of how random results work out. When fraud investigators are looking for bogus results, one of the things they look for is an absence of repeated numbers, like the prevalence of repeated won tosses. For example, an accountant faking the books would never enter a bill of $22,222. It looks too contrived. But random events include outcomes like winning a string of 19 of 25 guesses. It is the absence of repeated strings that triggers fraud detection. Or, from the NFL point of view, it would be an outcome where all team results were between 45% and 55% that would indicate fraud.

Assuming, that is, anyone in the league office understands math.

Urgent said:

Some basic probability:

The odds of this happening are 1 in 136.
There are 32 teams. That means there is a 1 in 4.25 chance it happens to any one team.
That is, there is about a 25% chance this happens in the league every 25 games.
There's about a 66% chance it happens once every six seasons.
It is not impossible. It is not unlikely. It is probable. It is more probable than not that this happens every six seasons.

Now, clearly, someone pulled a specific sample set - 19 of the last 25. Had the Patriots won the toss 26 games ago, they would have said 20 of the last 26. So we can surmise that the odds are slightly lesser, at 19 of the past 26. The pattern prior to the toss 26 games ago might appear more random. Although I guess one could argue that 25 games ago the Patriots learned to cheat the coin toss.

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Very good analysis. Though I'm not sure if the odds are exactly correct since the coin tosses for the 32 teams aren't independent (every time a team wins a toss, their opponent loses the toss). Conceptually, though, you're dead on. There are lots of teams so it's likely that once in a while, one of them will diverge from the expected value. And the 25 number (as opposed to 24, 26, etc.) was cherry-picked because it made the case better than the alternatives.

aluminum seats said:

Looks like it's pop culture day at Pats Fans!

There's a ton of good TV out there, should really take a look. TV is pretty much lapping the movies in terms of overall quality.

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Start with Breaking Bad

A Defiant Goose said:

Hey, f**k math to be honest

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Now we know who Goodell posts here as!

Folks, sad to say this story is actually picking up steam. I just read it in Boston's Prudential Tower's elevator news service. The headline read, "Another Patriots controversy brewing?"

Very frustrating.

Rob0729 said:

• Did each ref flip the coin the same way? If one refs flips the coin and the coin spins a lot in the air while another ref flips the coin and it doesn't spin at all, that will affect the probabilities of how the coin lands. The 50-50 probability is only when the coin is flipped exactly the same every time.

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Actually, if the coin is flipped exactly the same every time, the result will be exactly the same. In the study mentioned below, the researchers were able to build a machine that flipped the coin exactly the same way each time and make it come up the same every time:

Dynamical Bias in the Coin Toss

Coin tossing is a basic example of a random phenomenon. However, naturally tossed coins obey the laws of mechanics (we neglect air resistance) and their flight is determined by their initial conditions. Figures 1(a)–(d) show a coin tossing machine. The coin is placed on a spring, the spring is released by a ratchet, and the coin flips up doing a natural spin and lands in the cup. With careful adjustment, the coin started heads up always lands heads up—one hundred percent of the time. We conclude that coin tossing is “physics” not “random.”

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FourierSeries said:

Based on research, the probability would be 51% biased towards whichever side was up when the coin was thrown into the air. Also, if the coin spins in some way, there's as high as an 80% chance that it lands on on the heavier side.

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The research was based on the situation where the coin was caught and the spin was stopped. For a coin that's allowed to hit the floor and bounce, results are more random.

All of the studies cited above assumed the coin is caught in the hand without bouncing. An analysis of the effect of bouncing in coin tossing was suggested by Vulovic and Prange [37]. Following Keller, they assumed that the coin rotates about an axis through its plane. Thus, the phase space is ( ω , t ) as before. They hypothesized an explicit model for inelastic collisions that determines the coin’s eventual resting place. The resulting partitioning of ( ω , t ) space is surprisingly similar to Keller’s (see Figure 6). The preceding analysis (as illustrated in Figure 6) shows a reasonably fine partitioning of the phase space. In these regions, Vulovic and Prange showed that bouncing causes a fractal structure to appear in regions far from zero. Then, bouncing appreciably enhances randomness.

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In the NFL case, the coin is allowed to hit the turf. It probably bounces less than for a hard floor, but I would guess that the results are closer to random.

Hammer of Thor said:

Folks, sad to say this story is actually picking up steam. I just read it in Boston's Prudential Tower's elevator news service. The headline read, "Another Patriots controversy brewing?"

Very frustrating.

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??? How can that be? Its was satire.

Where is that little video with Jerry Seindfeld in the theater kinda giving up?

bbobbo said:

Actually, if the coin is flipped exactly the same every time, the result will be exactly the same. In the study mentioned below, the researchers were able to build a machine that flipped the coin exactly the same way each time and make it come up the same every time:

Dynamical Bias in the Coin Toss




The research was based on the situation where the coin was caught and the spin was stopped. For a coin that's allowed to hit the floor and bounce, results are more random.



In the NFL case, the coin is allowed to hit the turf. It probably bounces less than for a hard floor, but I would guess that the results are closer to random.

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Exactly, which is why I stated that scenario is unlikely, since the coin hits the turf.

The odds should be fairly close to 50/50.

bbobbo said:

The research was based on the situation where the coin was caught and the spin was stopped. For a coin that's allowed to hit the floor and bounce, results are more random.

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This is know by everyone who ever spent 5 minutes flipping a coin.

bbobbo said:

Actually, if the coin is flipped exactly the same every time, the result will be exactly the same. In the study mentioned below, the researchers were able to build a machine that flipped the coin exactly the same way each time and make it come up the same every time:

Dynamical Bias in the Coin Toss.

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Good stuff. At the end of the day, except for quantum mechanics, nothing is truly a random event (and I'm still not convinced it truly is the case there either, just doesn't sit right with me). It's just so hard to know all of the exact variables and inputs to be able to predict with certainty. It so happens that the coin flip result tends to mimic a 50/50 binomial process so it's an extremely accurate model to use for this in general. But as this shows, control and know all the conditions, the result can be predicted.

Pujo said:

None of those are relevant to the odds of the flip. The odds are that a fair coin thrown fairly lands on heads are always 50/50. That doesn't mean that you will have an even number of heads and tails, but over a large number of throws (much larger than 25) you'll be damn close. If you're not just counting heads but instead are counting "successes" (the statistical term), which we'll define as "whatever the Pats called as the coin was thrown", the odds are still 50/50, regardless of whether the Pats called heads, tails, there's wind, rain, etc. Even though those do affect the toss, they're not predictable (given a fair coin and a fair toss) by the team making the call. Obviously if you could perfectly model the dynamics (wind, rain, etc.,) then you could predict the coin toss perfectly, but AFAIK, nobody's managed that, and unknown variables that work symmetrically (i.e., just as likely to cause an otherwise heads-falling coin to become tails as they are to cause a tails-falling coin to become heads) don't affect the odds.

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Absolutely. Unless variables have a material impact on either a) the frequency at which one side will come up or b) the predictability of the flip, they don't need to be considered.

Bruins29 said:

??? How can that be? Its was satire.

Where is that little video with Jerry Seindfeld in the theater kinda giving up?

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You've heard about how Onion news stories were passed off as real in the media from time to time, right? This article may have been satire, but sadly someone decided to take it seriously....

Hammer of Thor said:

Good stuff. At the end of the day, except for quantum mechanics, nothing is truly a random event (and I'm still not convinced it truly is the case there either, just doesn't sit right with me). It's just so hard to know all of the exact variables and inputs to be able to predict with certainty. It so happens that the coin flip result tends to mimic a 50/50 binomial process so it's an extremely accurate model to use for this in general. But as this shows, control and know all the conditions, the result can be predicted.

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Let's not go all butterfly effect in here. Most things are purely random, or close enough to it that any difference is meaningless.

I was watching the Dolphins game in a restaurant in cinncinatti and when ghost kicked the field goal that curved left through the uprights I heard a guy at a table behind me say the patriots had a magnet in the ball and energized the goalposts!

Being a Patriots fan is great.. not only do you get to watch the team dominate, you also learn about physics, statistics, and how clock management and headset communication work in the NFL.

Lavictoire said:

I was watching the Dolphins game in a restaurant in cinncinatti and when ghost kicked the field goal that curved left through the uprights I heard a guy at a table behind me say the patriots had a magnet in the ball and energized the goalposts!

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Send it to cbs sports. They will make a headline out of it.