LeBron asks: What are the chances of making 10 free throws in a row? LeBron James asks Sal how to determine the probability of making 10 free throws in a row
LeBron asks: What are the chances of making 10 free throws in a row?
- 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
- 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
- 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 multiplications.
- 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.
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