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## 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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# Three-pointer vs free-throw probability

AP.STATS:

VAR‑4 (EU)

, VAR‑4.E (LO)

, VAR‑4.E.1 (EK)

, VAR‑4.E.2 (EK)

Our friend and Cleveland Cavalier, LeBron James, asks Sal if there's a high probability of making three free throws in a row or one three-pointer. Before solving the problem, jot down what you think the answer will be! Created by Sal Khan and LeBron James.

## Video transcript

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.