FrontBurner's Tim Rogers made a bet during Thursday's Mavs/Heat NBA Finals game. He won, but got to wondering if he had made a smart bet.
"I need help from someone who has a bigger math brain than I do. Last night, as Dirk was taking some free throws, I made a bet that he wouldn’t miss two in a row, and I gave 1,000-to-1 odds. I won a dollar. But was that a smart bet?"
Dirk Nowitzki’s career FT% is 88%. For 2010-2011 it’s 89%. That means he misses about 1 in 10. So, the odds of missing two in a row is 1/10 x 1/10 = 1/100. Giving 100-1 odds is fair. Giving 1000-1 is a dumb bet.
But Tim Rogers didn't say when he made his bet. If it was right after Dirk missed one free throw, then the odds of his missing his next one is right back to 1 in 10. Don’t give more than 10-1 odds in that case.
Other readers calculated slightly different odds based on using different history, say, just this year's playoffs (in which Dirk is shooting better than his career 88%). But the basic analysis stands.
Then, Tim Rogers clarified his question, saying that he made his bet after Dirk missed the first of two free throws. Tim Rogers explains his reasons for giving 1000-1 odds in that case:
"But this isn’t COMPLETELY driven by probability, is it? I’m thinking that if you looked at a large sample group -- all of Dirk’s free throws for a season say -- two consecutive misses would be rarer than his shooting percentage suggests it ought to be. Because if he misses the first one, he’s so awesome that he’ll adjust and make the second."
He's right. Shooting free throws is not exactly like, say, flipping a coin. In the case of coin flips, each flip is independent of the last. But with free throws, you could hypothesize that if he misses his first free throw, he’ll adjust on this next one, increasing his success rate. Or maybe he missed the first because the foul was a poke in eye, interfering with his natural ability, thus increasing the chances he’ll miss two in a row. Only by examining the history, shot by shot, could you tell if there’s anything to either hypothesis. Or whether each free throw attempt can be treated as if it's independent of the last.
So, let's look at the game-by-game, shot-by-shot, history. In the 2009-2010 season (the most recent available on BasketballGeek.com), Dirk shot 586 free throws, missing 50 for a 91.5% average. After missing one free throw, he made his next free throw 45 times, for a 90% average. It would appear that missing a free throw did not affect his average either positively (e.g, he made an adjustment) or negatively (e.g., a poke in the eye). Each free throw can be considered an independent event.
But hold on, you say. How many of those missed 50 free throws were on the front end of shooting a pair of free throws? The second free throw in those cases would more likely be affected by what happens on the first free throw than a case, for example, where Dirk misses the second free throw, then goes two or three minutes before getting fouled again and shooting his next free throw.
In the 2009-2010 season, when shooting two free throws, Dirk missed the first one 28 times. When he did, he also missed the second one 2 times. So, once he's missed one, you can give 26-2 or 13-1 odds against him missing the second, also. That's better shooting than the 10-1 odds that his 91% FT average for that season might suggest. So, maybe the sample size is too small, or maybe he really does adjust on that second shot and improve his chances, but if so, it's still nowhere near enough to give somebody 1000-1 odds.
Bottom line, Tim was letting his man crush on Dirk affect his bet.