In addition to electronics assembly productivity, solder paste and solder preform applications, and other electronics assembly concerns, many of you know that I teach statistics and have been a life-long math nerd. So, I was intrigued when good friend Rick Short, alerted me to the New Orleans Saints football team being 0-11 for coin tosses this season. One naturally asks, how unusual is this result?
Assuming a fair coin, there is a 50% chance of winning or losing on each flip. The chances of losing two times in a row is 0.5 x 0.5 = 0.25. The chances of losing 11 times in a row, in the first 11 tosses, is 0.5^11= 0.00048828125. Or about 2000 to 1 ( 1/0.00048828125 = 2048) as the article points out. Wow!, seems unusual. However, there are 32 teams, and it has only happened to one team. So what is the likelihood of this happening to one team this year? This calculation is a little more tricky. The easiest way to calculate it is to ask the question, what is the likelihood of this event not happening to any team?
The chances of not losing 11 in a row, in the first 11 tosses, for any team is 1-0.00048828125 =0.99951171875. So the chances that none of the 32 teams would lose 11 coin tosses in a row is 0.99951171875^32=0.98449268023. The chances that at least one team might lose 11 in a row is 1 minus this number or 1-0.98449268023= 0.015507319766 or about 1.55%. This number is still quite low. But what if we looked a 50 seasons?
The chances that 50 seasons would go by and no team would lose the first 11 coin tosses in a row is 0.98449268023^50 = 0.45774601688. So in 50 seasons, with a 32 team league, the chances are 45.77% no team would lose the first 11 coin tosses in a row, or 54.23% that at least one team would. Since these odds are close to 50/50, in fifty years, the Saint coin toss loss string is a 50 year event.
The article goes on to state:
“And while the Saints are 7-3 and lead the NFC South despite coming up short every single time on what should be a 50-50 proposition, coin-toss statistics — yes, they do exist — show that the NFL team that won the pregame flip wound up winning 52.1 percent of the time through Week 10 this season, according to STATS LLC.
That’s about the same as the 52.6 percent that STATS shows for coin-toss “victories” matching up with game victories since the start of the 2008 season, when the NFL changed the rules to allow the team that wins the toss to defer its choice until the second half.”
If there is interest, I will see I can calculate the statistical significance of this apparent coin toss win 2.6% advantage. My guesstimate is that the difference is quite statistically significant.
Note: Some readers may ask why I have used so many decimal places in my answers. Experience has taught me that when you are taking numbers to very high powers (the 32nd and then the 50th) that rounding errors can be great. In addition, noticed that I often said the "first 11 coin tosses." The odds would be higher to get eleven in a row out of a larger number of tosses, say 16.