If you were to bet 25 coin flips ago that someone would win 19 of 25 flips of a fair coin, then indeed the odds would be dissuading low (or the payout would be enticingly high). That the Patriots won 19 of 25 coin flips after the 25th flip, though, is meaningless. If the coin is fair, then each flip is an independent 50/50 proposition. "Independent" is key. If it's heads on one flip, it doesn't follow that tails must follow on the next. Heads could easy be realized on the next flip and many flips after. Such a trend provides no evidence of cheating or a conspiracy.
Few sports fans, and just as few analysts, realize that the 50/50 balancing act applies only to large numbers. After 10,000 flips, you should expect to see something approximating 50/50. Within those 10,000 flips,you should also expect to see streaks of heads and tails. A random slice of flips reveals no diabolical trend. Any small sample of flips could suggest an unfair coin or a cheater to the aggrieved, though nothing nefarious has occurred.
Winners and losers in an NFL coin flip follow a similar pattern. Winning 19 of 25 coin flips of 10,000 tells you nothing. A thousand flips from now, the Patriots might lose 19 of 25 flips, and then the New England faithful are pounding their chest about a different conspiracy.
Coins are one thing, humans are another. NFL head coaches are frequently criticized for play-calling conservatism. Punting on fourth-and-one is a recurrent example: Statistics show that odds overwhelmingly favor success when going for it on fourth-and-one: Going for it on fourth-and-one raises the probability of accumulating points during a game, regardless of field position.Despite the odd being in their favor, few NFL coaches will go for it unless they're within their own 45-yard line or they're losing in the waning minutes of the game.
With the punt-or-go-for-it scenario, we're dealing with probabilities based on the same law of large numbers as the coin flip. We're also dealing with another element -- humans. It's possible to posit average behavior for particles -- like those that comprise a fair coin -- that are hypothetically identical in structure and unchanging over time. A football game is a different matter; it's the domain of humans, who are undeniably individual and capriciously changeable.Because of the human factor, probabilities arrived vetting the past 10,000 games will tell you nothing about the expected probabilities of success on any particular play.
But let's say that humans are identical and unchangeable, the probabilities are still derived from large numbers. Head coaches lack the luxury of large numbers. If the head coach goes for it on fourth-and-one, the calculated probabilities might show he maximizes the probability of gaining points during a game.If the head coach fails, he can try to enlighten the owner with the statistics that show success is inevitable.
And if the head coach goes for fourth-and-one four-consecutive times and fails four-consecutive times,perhaps his new employer will be more receptive to the statistical probabilities.