Three-pointer vs free-throw probability

LEBRON JAMES: Hey, everybody. LeBron here. Got another quick brain teaser for you. Do I have a better odds of making 3 free throws in a row or 1 three-pointer? Here's my friend, Sal, with the answer. SAL: Excellent question, LeBron. But before I answer it, I want to point out an interesting trend related to your question that I just dug up. This is from the New York Times, October, 2009. So it's a couple of years old, but it's really interesting. It shows that since three-pointers were first introduced-- they were first introduced in the 1979 to 1980 season-- that three-pointers have become more and more frequent. So what they are showing here is the average attempts per team, season by season. And it looks like there's just a steady upward trend here related to our question. There's a couple of anomalies here. And the ones that really jump out are these three seasons in the late '90s. And that's because the actual three-pointer line was pulled in to essentially get higher scoring games. So people attempted more, but then it was put back to where it was originally. This was a shortened season. And I'm not really sure what happened in this, what is this, the 2001, 2002 season here. But it's something to think about. There is just this trend that more and more people are taking three-pointers. Now, with that out of the way, let's think about your actual question. And to answer it, I dug up your stats right over here from nba.com. And we'll use your career stats. So we want to compare 3 free throws to a three-pointer. So right over here, we have your three-pointer percentage, and this is in your career. And I'll round it to the nearest hundredths, so it looks like it's about 33%. So your three-point percentage is-- we'll just call it 33%. And then your free throw percentage, your career free throw percentage-- so this is free throw percentage, and this is in your career. We'll round to the nearest hundredths. So we'll round up right over here. That gets us to right at about 75%. So clearly looking at these numbers right over here, you're much more likely to make a given free throw than making a given three-pointer. You're more than twice as likely to make a free throw. But that's not what you asked. You asked what about 3 free throws in a row. And so what we'll do is we'll do an analysis very similar to the last time when we asked about 10 free throws in a row. So let's think about the first free throw. So free throw number one. If we imagined a gazillion-- a billion-- a gazillion's not a real number. If we imagined a billion LeBron Jameses, identical LeBron Jameses all taking that first free throw, we would expect, on average, that 75% would make that first free throw. So 75% is 3/4. So 75% would make that first free throw. And 25%, we would expect, on average-- wouldn't always be the case, but this is what we would expect. 25% would miss that first free throw. Now, let's go to the second free throw, free throw number two. And we only care about the LeBron Jameses that keep making their free throws. So let's think about of the 75% that made that first one. Some of the 25% might make that second one and then maybe even the third one. But let's just think about the ones that made the first one. Of the ones that made the first one, we would expect 75% of them to make the second one. So 75% of the 75%, that's 1/2 of the 75%. That's about 75% of the 75% would make that second free throw and the first free throw. So now we have-- this is going to be 75% times 75%. And, of course, there's other combinations out here where someone's missed at least one of the free throws. Now, let's go to the third free throw, free throw number three. What percentage of these LeBron Jameses right here will make the third free throw? Well 75% of these will make the third free throw. So 75% of this number. So let me just draw it visually. That's 1/2. That's about 75% of that number. They will make the third one, as well. And so this is 75% of this number, which is 75% of 75%. This is how many LeBron Jameses are going to make all 3 of the free throws. And, once again, we can write this as-- we could either multiply it out or we can just write this as 75% to the third power, which is the same thing as 75% literally means 75 per 100. Same thing as 75 over 100 to the third power, which is the same thing as 0.75 to the third power. And so let's calculate it. Get the calculator out. And actually, let me show you we get the same result. We can write 0.75 times 0.75-- and on this calculator, that little snowflake looking thing, it means multiplication-- times 0.75. And then we get 0.42. I'll round to the nearest hundredths. And that's the same thing we would get if we got 0.75 to the third power. Once again, 0.42. So let me write that. So this gets us to approximately 0.42, which is the same thing as 42%. So your probability of making 3 free throws in a row is 42%, which is still higher than making 1 three-pointer. So I'll leave you there, but I want the people who are watching this video to think about what would happen if the numbers were different? Or maybe look up some NBA players, or maybe some college players, and figure out and compare the probability of making 3 free throws to a three-pointer, and see if you can find any players where their probability of making a three-pointer is actually higher than making 3 free throws in a row.