I'm about to get to the segment of my research methods course covering probability, and I have a new favorite example of the cumulative probability of independent events. Too bad I can't use it in class, given that football is a rather culturally biased source for anecdotes.
Super Bowl viewers may recall that despite a furious New England comeback, the Atlanta Falcons were leading 28-20 with four minutes remaining and in a dominant position – first down at New England's 22 yard line. From here, literally all the Falcons had to do was fall down three times (which would either force New England to spend their timeouts or run 2 precious minutes off the clock), kick a simple 40-yard field goal, and return the ball to New England down 11 points with little time left. In other words, a Falcon victory was virtually guaranteed.
How guaranteed? Well, consider if Atlanta ran the simplest of plays on 1st, 2nd, and 3rd down: a run straight up the middle by very excellent running back Devonta Freeman. It didn't matter if he gained yards or not. Running the clock and, crucially, NOT fumbling the ball away were all that mattered. Fortunately for Atlanta, Freeman carried the ball 227 times and caught 54 passes this season with all of one fumble, so we can calculate his odds of running the play without fumbling as 1 – 1/(227+54), or 99.64%. To calculate the odds of running the play three times (independent events) without fumbling, we cube that figure, (0.9964)
On fourth down, Falcons kicker Matt Bryant would appear to kick a field goal of just under or over 40 yards. This season he made 95% (19/20) of kicks under 40 yards, and 97% (28/29) under 50. The 41 year old veteran has made 300 field goals in the NFL over 15 years, so presumably nerves wouldn't have drastically altered his performance. But for the sake of being conservative, let's say his odds were 92% (his total season average). 98.92% x 92% = 91%. In other words, by doing nothing but what was obvious, the Falcons had at least a 91% chance of taking an 11 point lead and essentially guaranteeing victory.
Instead, Atlanta got too cute. On 1st down Freeman carried the ball for a short loss. On second down, Atlanta passed for baffling reasons. QB Matt Ryan was sacked, losing 12 yards and making a potential field goal very long. Then they passed again, this time drawing a very obvious holding penalty and losing 10 more yards. Now, no field goal attempt was possible. The rest is history.
Consider what they did there. Leave aside for a moment the 30% chance of a pass being incomplete and stopping the clock, which would be bad (helping New England) but not fatal. QB Ryan had 1.3% of his passes intercepted, was sacked 6.5% of the time he attempted to pass, and the Falcons performed near the NFL average of an offensive penalty on 1 of every 10 plays (10%). The sum of those (17.8%) is the probability that something really, really bad could happen on a pass attempt. That leaves an 82.2% chance that things will be alright on a pass attempt. Counting Freeman's first down run (0.9964), we then multiply by (0.822) x (0.822) for the second and third down passes, giving us 67.3% probability of these three plays being run without "something bad" happening. Multiply that by Bryant's 92% chance of making a field goal, and we see that the plays Atlanta actually ran gave them only a 61.94% chance of getting that crucial 11-point lead.
NFL coaches may not be rocket scientists but most could tell you that 91% is greater than about 62%. And remember, 91% is an extremely low, conservative estimate.
We see this all the time in football; coaches get too fancy trying to "outsmart" the odds. When it works, they're praised for being Bold. But math is going to win more regularly than Guts or Boldness or anything else.