Free-throw probability

LEBRON JAMES: Hey, everybody. It's LeBron here. I got a quick brain teaser for you. What are the odds of making 10 free throws in a row? Here's my good friend, Sal, with the answer. SAL: That's a great question, LeBron. And I think the answer might surprise you. So I looked up your career free throw percentage, and you're right around 75%, which is a little bit higher than my free throw percentage. And one way to interpret that, if we have a million LeBron Jameses, as you can imagine, any large number of LeBron Jameses is taking a free throw. So let's say that this line represents all of the LeBron Jameses that take that first free throw. Let's call that free throw number one. We would expect, on average, that 75% of them would make that first free throw. So let me draw 75%. So this is about half way. This would be 25. This would get us to 75. So we would expect 75% of them would make that first free throw. 75%. And then the other 25% we would expect, on average, would miss that first free throw. Now, what we care about are the ones that keep making the free throws. We want 10 in a row. So let's just focus on the 75% that made the first one. Some of these 25% might make some free throws going forward, but we don't care about them anymore. They're kind of out of the game. So let's go to free throw number two. What percentage of the folks who made, of the LeBron Jameses, that made that first free throw, what percentage would we expect to make the second one? We're going to assume that whether or not you made the first one has no bearing on the probability of you making the second, that this continues to be the probability of a LeBron James making a given free throw. So we would expect 75% of these LeBron Jameses to also make the second one. So we're going to take 75% of 75%. So this is about half of that 75%. This would be a quarter. This would be 3/4, which is exactly 75%. So right over here. So this represents, of the ones that made the first one, how many also made the second one. So you could say the percentage of the LeBron Jameses that we would expect on average to make the first two free throws. So this length right over here is 75% of 75%, 75% of this 75% right over there. And I think you might begin to see a pattern emerging. Let's go to the third free throw. Free throw number three. So what percentage of these folks are going to make the third one? Well, 75% of them are going to make the third one. So 75% are going to make the third one. So what is this going to be? This is going to be 75% of this number, of this length, which is 75% of 75%. And if you were to go all the way to free throw number 10, and I think you see the pattern here, if we were to go all the way to free throw number 10-- so I'm just skipping a bunch. And we're going to get some very, very, very small fraction that have made all 10, it's essentially going to be 75% times 75% times 75% 10 times. 75% being multiplied repeatedly 10 times. So this is going to be what we're left off with. It's going to be 75% times 75% and let me copy and paste this so it just doesn't take forever. So copy and then paste it. So times out-- I'll put the multiplication signs later. So that's 4, that's 6, that's 8, and then that is 10 right over there and let me throw the multiplication signs in there. So times, times, times, times. So this little fraction that made all 10 of them is going to be equal to this value right over here 75%. So let's see. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 75% being repeatedly multiplied 10 times. Now this would obviously take me forever to do it by hand. And even on a calculator, if I were to punch all of this in, I might make a mistake. But lucky for us, there is a mathematical operator that is essentially a repeated multiplication, and that's taking an exponent. So another way of writing that right over there, we could write that as 75% to the 10th power, repeatedly multiplying 75% 10 times. These are the same expression. And 75%, the word percent literally means per 100. You might recognize the root word cent from things like century. 100 years in a century. 100 cents in $1. So this literally means per 100. So we could write this as 75 over 100 to the 10th power, which is the same thing as 0.75 to the 10th power. Now, let's get our calculator out and see what this evaluates to. So 0.75 to the 10th power gets us to 0.056, and I'll just round to the nearest hundredths. So if we round to the nearest hundredths, that gets us to 0.06. So this is roughly equal to, if we round to the nearest hundredths, 0.06, which is equal to roughly, when we round, a 6% probability of making 10 free throws in a row. Which even though you have quite a high free throw percentage, this is not that high of a probability. It's a little bit better than a 1 in 20 chance. Now, what I want to throw out there, for everyone else watching this, is to think about how we can make a general statement about anybody. If anybody has some free throw percentage, and they want to say, what's the probability of making 10 in a row? How can we say that? Well, I think you saw the pattern right over here. The probability of making-- let's call it n where n is a number of free throws we care about-- n free throws in a row for somebody. And we're not just talking about LeBron here. It's going to be their free throw percentage-- in this case, LeBron's was 75%-- to the number of free throws that we want to get in a row. So to the nth power. So for example, you might want to play along with their own free throw percentage. If your free throw percentage, let's say it's 60%, which is the same thing as 0.6. So let's say you have a 60% free throw percentage, and you want to see your probability of getting 5 in a row, you would take that to the fifth power. And you'd get what looks like, if you round to the nearest hundredths, it would be about 8%. So I encourage you to try this with different free throw percentages and different numbers of free throws that you're attempting to get in a row.