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Course: Statistics and probability > Unit 9

Lesson 5: Binomial random variables
Math
Statistics and probability
Random variables
Binomial random variables
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Binomial probability (basic)

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Problem 1: Building intuition with free-throws

Steph makes 90% of the free-throws she attempts. She is going to shoot 3 free-throws. Assume that the results of free-throws are independent from each other.
She wants to find the probability that she makes exactly 2 of the 3 free-throws.
To think about this problem, let's break it up into smaller parts.
problem A
If she makes 2 of the free-throws, how many free-throws does that mean she needs to miss?
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problem b
Find the probability that she makes her first 2 free-throws and misses her third free-throw.
Round your answer to the nearest hundredth if necessary.
P(make, make, miss)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

problem c
"Make, make, miss" isn't the only way Steph can make make 2 free-throws in 3 attempts.
Find the probability that she makes her first free-throw, then misses the second, and then makes her third free-throw.
Round your answer to the nearest hundredth if necessary.
P(make, miss, make)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

problem d
Steph could also make 2 free-throws if her results are "miss, make, make".
Find the probability that she misses her first free-throw and makes her next 2 free-throws.
Round your answer to the nearest hundredth if necessary.
P(miss, make, make)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

problem E
Use the combination formula to verify that these 3 ways represent every way we can arrange 2 makes in 3 attempts.
nCk=n!(n−k)!⋅k!
3C2=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
ways

problem f
Now put it all together to find the probability that she makes exactly 2 of the 3 free-throws.
Round your answer to the nearest hundredth if necessary.
P(makes 2 of 3 free throws)=P(S S F)+P(S F S)+P(F S S)
P(makes 2 of 3 free throws)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Generalizing from Problem 1: Building a formula for future use

We saw in Problem 1 that different orders of the same outcome each had the same probability.
We can build a formula for this type of problem, which is called a binomial setting. A binomial probability problem has these features:
  • a set number of trials (n)
  • each trial can be classified as a "success" or "failure"
  • the probability of success (p) is the same for each trial
  • results from each trial are independent from each other
Here's a summary of our general strategy for binomial probability:
P(# of successesgetting exactly some)=(arrangements# of)⋅(of successprobability)(successes# of)⋅(of failureprobability)(failures# of)
Using the example from Problem 1:
  • n=3 free-throws
  • each free-throw is a "make" (success) or a "miss" (failure)
  • probability she makes a free-throw is p=0.90
  • assume free-throws are independent
P(makes 2 of 3 free throws)=3C2⋅(0.90)2⋅(0.10)1=3⋅0.81⋅0.10=3⋅0.081=0.243

In general...

P(exactly k successes)=nCk⋅pk⋅(1−p)n−k
Try using these strategies to solve another problem.

Problem 2

Steph's little brother Luke only has a 20% chance of making a free-throw. He is going to shoot 4 free-throws.
What is the probability that he makes exactly 2 of the 4 free throws?
P(exactly 2 makes)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Challenge problem

Steph promises to buy Luke ice cream if he makes 3 or more of his 4 free-throws.
What is the probability that he makes 3 or more of the 4 free throws?
P(3 or more makes)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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