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Free-throw probability

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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. 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 are right at 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 James, and you can imagine any large number of LeBron James is taking a free throw. Let's say, this line represents all of the LeBron James' that take that first free-throw Let's call that free throw number 1 We would expect, on average, that 75% of them would make that first free-throw. So, let me draw 75%, so this is about halfway. So this would be 25, This would get us for 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 ten 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 are kind of out-of-the-game! So, let's go to free-throw number two. Free throw ... number two. What percentage of the folks who made of the Lebron James-es, that made the first free-throw, what percentage would we expect to make the second one? And we're going to assume, whether or not you made the first one, has no bearing on the probability of you on 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 James' to also make the second one. We are going to take 75 percent of 75 percent. So this is about half of that 75%; this would be a quarter, this would be three fourths, which is about 75%, which is exactly 75%. So right over here. This represents the ones that made the first one, how many also made the second one. So you could say that the percentage of the Lebron James-es that we would expect on average to make the first two free-throws. This is...This is the length right over here is 75 percent of 75%. 75% of this 75% right over there. And I think you might began to see a pattern emerging, Let's go to the 3rd free-throw: free throw number 3. So what percentage of these folks are going to make the third one? Well, 75% of them are going to make the 3rd one. So, 75% of them are going to make the 3rd one. What is this going to be? This is going to be 75 percent, 75 percent of this number, of this length, which is 75% of 75%. And it if you would go all the way to free throw #10, and I think you see the pattern here, (if we are going all the way to Free Throw #10), so I am just skipping a bunch, we are going to get some very, very, very small fraction that had made all ten, is 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 have left off with, this is going to be 75%, times 75% (and let me copy and paste this) (so it doesn't take forever.) (So... copy, and then paste it. So times out) (I will put the multiplication signs on later) (that's four... that's six... that's eight..) (...and then: that IS ten. Right over there,) (let me throw the multiplication signs in there, so) (times, times, times, times...) So this little fraction that made all ten of them is going to be equaled to this value right over here. 75 of... let's see: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10: 75 percent being repeatedly multiplied 10 times. Now, this would obviously be taken me forever to do it by hand. And even on a calculator, if I would have punched all of these in, I might made mistake(s). But lucky for us: there is a mathematical operator that is essentially repeated multiplication. And that's taking an exponent. So another way- another way of writing that, (right over there), we could write that as: 75 percent to the... tenth power, repeatedly multiplying 75% ten times- these are the SAME expression(s). And 75 percent, the word "per-cent", literally means "per hundred." You might recognize the root word, "cent," from things like "century," 100 years in a century, 100 cents in a dollar, so this literally means "per hundred." So we could write this as "75 over 100 to the 10th power," which is the same thing as 0.75 to the tenth power. And let's get out calculator out and see what this evaluates to. So, 0.75 to the... tenth power, gets us to: .056, and I'll just round to the nearest hundreds, so if we'd round to the nearest hundreds, that gets us to .06. So this is roughly equaled to (if we round to the nearest hundreds): 0.06 which is equaled to, roughly when we round, a 6 percent probability of making ten 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 one-in-twenty chance. :D 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 that anybody has some free throw percentages, and wants to say: what's the probability of making 10 in a row? How can we say that? Well, I think we saw the pattern right over here. The probability... of making, let's call it "n," where "n" is the number of free-throws we care about, n free-throws in a row, for somebody, and we are not just talking about LeBron here, is going to be their free throw percentage, and this case, Lebron's is 75%, to the number of free throws we want to get in a row. So to the, n-th power. For example: you might want to play around with your own free throw percentages. if your free-throw percentage is, let's say, 60 percent, which is the same thing as .6. So let's say you have a 60% free-throw percentage, and you want to see the probability of getting five in a row, you would take that to the fifth power. And you'd get- what looks like, if you round it to the nearest hundredth, would be about 8%. So I encourage you to try this with different free-throw percentages, and different numbers of free throws that you are attempting to get in a row.