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### Course: Statistics and probability > Unit 7

Lesson 7: Multiplication rule for independent events- Sample spaces for compound events
- Sample spaces for compound events
- Compound probability of independent events
- Probability of a compound event
- "At least one" probability with coin flipping
- Free-throw probability
- Three-pointer vs free-throw probability
- Probability without equally likely events
- Independent events example: test taking
- Die rolling probability with independent events
- Independent probability
- Probabilities of compound events
- Probabilities involving "at least one" success
- Probability of "at least one" success

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# Free-throw probability

Our friend and Cleveland Cavalier, LeBron James, asks Sal how to determine the probability of making 10 free throws in a row. Hint: the answer is surprising! Created by Sal Khan and LeBron James.

## Want to join the conversation?

- Wouldn't making the first free throw increase his FT%? Say his initial FT% was 75% and he made the first shot, increasing his FT% to 76%, you would multiply 75% by 76%, rather than doing 75%^2.(277 votes)
- We are assuming that 75% is his true free-throw percentage (the true probability of him making a free throw) and doesn't change from shot to shot. That number cannot be directly known, so we estimate it by looking at his history of making free throws. This is a reasonable thing to do because LeBron has taken thousands of FTs. His true ft% won't change from shot to shot. Our estimate of it might in the future, but for the sake of a question like this one, we assume that our current estimate is the true FT% for the entire experiment.

You are touching on a very important point in statistics. For the most part, we can only estimate a true parameter by looking at samples.(552 votes)

- I"m confused as to why we are taking 75% of 75% if the free throws are independent events.(12 votes)
- I also had a hard time grasping why he was taking 75% of 75% and so on. For some reason none of the other answers to this question in the comments clicked for me. Then I realized this example is exactly the same as figuring out the probability of flipping a coin 3 times and getting HEADS all three times. The math is the same, but Sal is presenting a different way of looking at the problem.

Ok, let's look at if we were calculating the probability of flipping a coin 3 times and getting HEADS all 3 times (flipping HEADS 3 times IN A ROW).

P(HHH) is the same as P(H) * P(H) * P(H)

When flipping a coin, the probability of flipping HEADS is 1/2. So then...

P(HHH) = P(H) * P(H) * P(H) = 1/2 * 1/2 * 1/2 which is the same as (1/2) ^ 3

If we go back to the free throw example, if F represents making a free throw, our problem would look like this:

P(FFFFFFFFFF) = P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F)

The probability of Lebron making a free throw is 75% or .75, so then...

P(FFFFFFFFFF) = P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) * P(F) = .75 ^ 10

I hope this helps others with the same question. Forgive me if you didn't need the breakdown as I have done above, but for me, that was how I was able to make sense of the problem. If my explanation does not help, I suggest re-watching the Coin Flipping Probability video as that was helpful for me.(41 votes)

- How much greater would it be if it were 90%?(18 votes)
- 0.3486784401 rounded to the nearest hundredth=0.35x100=35%(5 votes)

- This solution assumes that each free throw is an independent event. Once Lebron makes a shot I would argue that his probability of sinking the second one is higher than 75%. He knows the feel of the ball, what the arc should look like etc. I think the question that Sal answers is what is the probability of Lebron sinking his first free throw shot in 10 consecutive games which would be a more independent set of events.(9 votes)
- You are saying you think Lebron gets a "hot hand". It's an interesting hypothesis, but don't be too confident in it:

http://en.wikipedia.org/wiki/Hot-hand_fallacy(4 votes)

- if you do 10 free throws and your hands get tired doesn't that mess up the calculations(6 votes)
- This calculation is based on the assumption that you will have equal chances of success in each throw. If later throws have lower chances of success (because of tired hands or otherwise), the calculation will be different, but similar method can still be applied.

In short, yes, the calculation will not correctly represent the chances of making 10 free throws in a row.(3 votes)

- What is the significance of the parentheses at4:35?(3 votes)
- Putting the base number within parentheses is a common convention with exponents especially when we have a unit or a fraction. This is more important with other units because 10cm^2 is different from (10cm)^2. The first one is ten centimeters squared; the second is one hundred centimeters squared.(8 votes)

- at1:53, how could you multiply 75% *75%?(2 votes)
- 75% can be treated as 0.75. So you can simply multiply 0.75 by 0.75 to get the answer.(9 votes)

- How does this work if as you are shooting you are atcually increasing your accuracy?(3 votes)
- The percentage calculated by Sal is based on Lebron's
**average**percent of making a free throw. He is not taking into factor that Lebron is getting better with each shot, just using plain math. Now, to find out the percentage given that Lebron is getting better with each shot, you would have to find out how much better he is getting with each shot, take into factor age, etc... Honestly, I think if you try to figure that out, it's making the original problem more complicated than it should be.(6 votes)

- Why is the probability of missing 10 in a row, if the chance of making it is 75%

(1 - 75%)^10

and not

1 - (75%)^10 ?

or is it the same thing?(2 votes)- (1-0.75)^10 = (0.25)^10 correct

Both those expressions are not the same

1-(0.75)^10 = 0.94 ...

(1-0.75)^10 = 0.00000095 ...

totally makes a difference(8 votes)

- how come a 60% free throw percentage has a greater chance of making 10 in a row than 75%?(3 votes)
- It doesn't; the calculations he performed in the video were:

P(10 free throws in a row @ 75%) ≈ 6%

P(**5**free throws in a row @ 60%) ≈ 8%(3 votes)

## Video transcript

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.