We are 21 races into the 2013 season, and many big name drivers are still looking for their first win:
- Clint Bowyer
- Dale Earnhardt, Jr.
- Jeff Gordon
- Brad Keselowski
- Denny Hamlin
Are they winless because of bad driving and team troubles, or simply due to chance?
Today we look at the chances these drivers would be winless anyway, simply due to randomness and luck. It’s possible a driver and his team could do everything right and still come up short. In the same way a coin flip doesn't always come up heads, random chance plays a big factor in determining outcomes, regardless of one’s effort.
We answer this question by going back to our trusty friend, the binomial distribution, which we used two months ago to calculate the chances of Denny Hamlin and others getting two wins before the Chase.
The binomial distribution says how likely it is for a driver to win X races in a given period of time, based on his career winning percentage. This stat gives us an expected range of wins for each driver. It also says how many wins are outside the norm, either too many or too few for a driver to reasonably get.
Let’s start with a simple example. Jimmie Johnson has won 64 races in his career of 420 starts, a career winning percentage of 15%. Applying the formula, we should expect his wins per season to be distributed liked this:
I’ve highlighted the fields with a > 10% expectation. The highest-probability outcomes for Johnson are in the 3-7 win range, accounting for 76% of the most likely results. The math works out, as 9 of Johnson's 12 fulltime seasons (or 75%) have resulted in 3-7 wins. We now see Johnson's fantastic 10-win season in 2007 had a slim 2% chance of happening. Clearly he had good luck on his side, including equipment, strategy, and avoiding crashes. Even for an elite driver like Johnson, winning 10 races should only happen once every 50 years. He needed everything to go right to make it happen that year.
Similar reasoning applies to other extraordinary NASCAR performances. Consider Tony Stewart’s 2011 Chase, when he won 5 out of 10 races. The numbers suggest he only had a 0.1% of doing that by random chance (put another way, it’s a staggering once-in-thousand years opportunity). Again, he had many factors swing in his favor: the right setups, equipment, and a healthy dose of luck.
Let’s look at the expected wins so far in 2013, after 21 races, for the top 20 drivers in points:
The red boxes highlight the most likely outcomes.
- Jimmie Johnson has 4 wins already this year, which is right in line with his career performance.
- Clint Bowyer had a 54% chance of winning 0 races by this point in the season. That he has 0 wins is perfectly sensible: it's his most likely outcome.
- The same reasoning applies to Dale Earnhardt, Jr. He had a 44% chance of earning 0 wins in 21 races, higher than any other outcome.
- For Brad Keselowski, getting 0 wins so far had a 26% likelihood. Even if he didn't have to deal with the pressures of being a defending champion, switching manufacturers, and adjusting to the Gen-6 Car, it's perfectly possible he would have been winless anyway.
- In Jeff Gordon’s case, having 0 wins is a clear underachievement based on his career stats. His most likely outcomes would include winning anywhere from 1 to 4 races. His performance so far suggests this is not due to random chance, but a true decline at this late stage of his career, the natural dip that occurs as drivers move past their prime.
- Contrast that with Matt Kenseth, the only driver who has many more wins than expected. His 4 wins puts him way above his most likely range of 0-2 wins. In fact, his career track record suggests he only had a 2% chance of winning 4 races so far. What changed? Kenseth switched over to Joe Gibbs Racing this year – clearly the move has made a significant and measurable difference for him.
- On the other hand, Kenseth's JGR teammate, Denny Hamlin, has had a rough season, including sitting out several races due to injury. For Hamlin's 17 starts this year, going winless had a 24% chance of happening in normal circumstances. He was just as likely to have 0 wins (24%) as to have 2 wins (25%). These numbers tell us even if everything were going normally for Hamlin, he might still be winless from randomness alone.
Now let's see the likelihood of each driver’s win potential for the final 15 races this season:
This table gives you a sense for the most common outcomes. For most drivers, the most likely results are winning 0 or 1 more race for the remainder of the year. Outperformance (i.e. results that fall past the red boxes) will come from a combination of positive factors all working together to benefit a lucky driver. For example, you might see Kyle Busch win 3 more races before the season ends (9% chance of happening).
Finally, we can combine the two tables to look at the expectations for a complete 36-race season.
Look for your favorite driver and see what their most likely outcomes are.
It would be fair to say that if you are a crew chief looking to save your job, or a driver looking for a sponsor, you can point to these stats to show that even if you had an “off year”, the results might be nothing more than random chance working against you. The public's definition of a “good year” versus a “bad year” may be too narrow, putting unrealistic expectations on everybody in the field.
We need to remember there is so much randomness that affects any given race, and sometimes drivers are just unlucky, no matter how carefully they and their team prepare. Just like in a bad coin toss, sometimes a driver ends up on the wrong side.
Let's also consider that a great season may not be indicative of how good a driver truly is, but rather a lucky streak that may not be matched again. For example, Brad Keselowski won 5 races in 2012. The numbers say he only had a 5% chance of doing that. Based on his career so far, he should perform this well once every 20 years, suggesting that luck was on his side last year. In fact, the numbers say he is more likely to win 0 races rather than 5. Perhaps his winless 2013 is his luck evening out from last year’s success. Maybe nobody is at fault: he and his team could have done everything right, and random chance would still cause this.
That being said, he still has a 61% chance of winning at least one race before the end of the season. The odds are in his favor to do it.