1. 6.3 (part I) Binomial Random Variables

2. What is a Binomial Random Variable?
The short version:
A process occurs over and over again (or at least could happen over and over)
Each time it is repeated, the result is independent of the previous time
The result can be defined as “success” or “failure”
Or “yes” or “no”
Some examples:
Tossing a coin, counting the number of heads
Spinning a roulette wheel, counting the number of times it lands on red
Counting the number of females born in a random sample of babies born
Notice the theme of “counting”

3. BINS
A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome occurs. The four conditions for a binomial setting are:
Binary: The possible outcomes can be classified as either “success” or “failure” (“yes” or “no”)
Independent: Trials must be independent from each other. Knowing the result of one trial must not have any effect of the result of any other trial
Number: The number of trials of the chance process must be fixed in advance (we must know how many trials we are going to do)
Success: On each trial, the probability of success must be the same

4. Binary
The binary requirement is not actually as restrictive as it seems
For example, a deck of cards has 4 different suits
So if I am interested in the suit of a randomly selected card, it doesn’t seem binary
Because there are 4 different possible outcomes
BUT, if I count the number of spades I get in n trials, then it is binary
Because getting a spade is a “success” and getting any other suit is a “failure”

5. Independent This is oftentimes the most confusing requirement
Take our card suits example from the previous slide
If we sample cards WITHOUT replacement, then it is NOT independent
Because knowing whether the previous card was a spade would affect the probability of the next one being a spade
If we sample cards WITH replacement, however, then it is independent (as long as we shuffle)
Now knowing what the previous card was does not give you any information about the next one

6. Independent (again) An example:
You stand in the hallway and observe whether each of the next 20 people that walk by you is female
This is binary because we can answer it with a “yes” or a “no”
What about independence?

7. Independent (again) An example:
You stand in the hallway and observe whether each of the next 20 people that walk by you is female
This is binary because we can answer it with a “yes” or a “no”
What about independence?
Probably not independent, since knowing whether the first person is female might give you some information about whether the next person will be female

8. Number of Trials This one is the easiest
Do we know how many trials there are going to be before we start?
If yes, then it meets the requirement
If not, then it doesn’t
GOOD: Jim shoots 20 free throws and counts how many he makes
BAD: Jim shoots free throws until his arm gets tired, and counts how many he makes

9. Success
The probability of a success has to stay the same for each trial
In some ways, similar to the “independent” requirement
Oftentimes, something that violates one will also violate the other as well
But not always
Let’s look at the shooting free throws example. Jim shoots 20 free throws and counts how many he makes
This seems like it is probably fine: the probability of making his 4th free throw should be the same as the probability of making his 15th free throw
Unless he hasn’t warmed up, so the first few might have lower probabilities. In this case, the requirement would NOT be met

10. An Example
Jim’s friend Katherine thinks that shooting free throws is boring, and she wants more variety. So she decides to shoot 20 shots from various distances on the floor, counting how many she makes
Is this binary?
Is this independent?
Do we know the number of trials in advance?
Does the probability stay the same?

11. An Example
Jim’s friend Katherine thinks that shooting free throws is boring, and she wants more variety. So she decides to shoot 20 shots from various distances on the floor, counting how many she makes
Is this binary? YES—makes it or misses it
Is this independent?
Do we know the number of trials in advance?
Does the probability stay the same?

12. An Example
Jim’s friend Katherine thinks that shooting free throws is boring, and she wants more variety. So she decides to shoot 20 shots from various distances on the floor, counting how many she makes
Is this binary? YES—makes it or misses it
Is this independent? Probably—unlikely that making one shot would make her more likely to make the next
Do we know the number of trials in advance?
Does the probability stay the same?

13. An Example
Jim’s friend Katherine thinks that shooting free throws is boring, and she wants more variety. So she decides to shoot 20 shots from various distances on the floor, counting how many she makes
Is this binary? YES—makes it or misses it
Is this independent? Probably—unlikely that making one shot would make her more likely to make the next
Do we know the number of trials in advance? YES; 20
Does the probability stay the same?

14. An Example
Jim’s friend Katherine thinks that shooting free throws is boring, and she wants more variety. So she decides to shoot 20 shots from various distances on the floor, counting how many she makes
Is this binary? YES—makes it or misses it
Is this independent? Probably—unlikely that making one shot would make her more likely to make the next
Do we know the number of trials in advance? YES; 20
Does the probability stay the same? NO! she is way more likely to make a shot from 4 feet away than a shot from half court

15. Now…what can we actually do with a binomial distribution?
Binomial distribution questions are among the lowest-scoring free response questions on the AP test
Mostly because people don’t realize that it is a binomial distribution question
So what we’ve done so far is actually really important— identifying whether it is a binomial setting
What this allows us to do is calculate the probability of getting a specific number of successes out of n trials

16. An Example
A couple is planning on having five children. For each of their children, the probability of the child having Type-O blood is The count (X) of their 5 children that have Type-O blood is a binomial random variable with n=5 trials and probability p=0.25. What is the probability that they have exactly 2 children with Type-O blood?
This could be written as “P(X=2)”
We define a “success” as having Type-O blood
“Failure” as having any other type of blood

17. Example (continued)
So if S is a success and F is a failure, we need two out of the 5 to be “S”
SSFFF (.25)(.25)(.75)(.75)(.75)=.02637
SFSFF (.25)(.75)(.25)(.75)(.75)=.02637
SFFSF (.25)(.75)(.75)(.25)(.75)=.02637
SFFFS (.25)(.75)(.75)(.75)(.26)=.02637
FSSFF =.02637
FSFSF =.02637
FSFFS =.02637
FFSSF =.02637
FFSFS =.02637
FFFSS =.02637

18. Example (continued)
Since there were 10 possible combinations, the probability of getting exactly 2 children with Type-O blood would be:
10*(.02637)=.2637
So we can actually calculate binomial probabilities without knowing anything about the binomial distribution
We just did this
However, there are easier ways
We can use the binomial probability formula
And, eventually, our calculator

19. Binomial Probability Formula
N= number of trials
K= number of successes
P=probability of a success on a given trial

20. Binomial Coefficient

21. Quick Binomial Coefficient example
If we want the binomial coefficient for a binomial setting with 20 trials, where we want to know the probability of exactly 11 successes
It would be
Or 20! (11!)(9!) =167960
Use factorial button on your calculator (Math—PROB)
20! 11! 20−11 !

22. Continuing the Example
Let’s say that p= 0.6
We can calculate the probability of 11 successes out of 20
𝑃 𝑋=11 = =0.1597

23. Back to the Blood Type Example
Let’s try it using the binomial probability formula
Before, we calculated the probability to be .2637
N=5
K=2
P=0.25
Let’s first do the binomial coefficient
5! (2!)(3!) =10

24. Back to the Blood Type Example
Let’s try it using the binomial probability formula
Before, we calculated the probability to be .2637
N=5
K=2
P=0.25
Now we do the rest of the formula
𝑃 𝑋=2 = =.2637

25. Using our Calculator “choose” notation (binomial coefficient)
Would be read as “11 choose 5”
11! 5! 11−5 ! =462
You can use the nCr function on your calculator
11𝐶5=462
Press 11, then nCr, then 5

26. Using our Calculator for Binomial Random Variables
Our calculator can also directly calculate binomial probabilities
Binompdf(n,p,k) computes the probability that X=k
Binomcdf(n,p,k) computes the probability that X≤k
Remember, n is the number of trials
P is the probability of success in any given trial

27. Let’s Try an Example
A football team has a 40% chance of winning each of their games. Over the course of a 12-game season, what is the probability that they win exactly 7 games?

28. Let’s Try an Example First question…is it a binomial setting?
Binary? Yes—either win or lose
Independent? probably—we are assuming that whether they win one game doesn’t affect the next
Number? Yes, we know the number of trials (12 games)
Success? In real-life, probably not, because they would play some easier teams and some harder teams. But in this example, it specifically says that they have a 40% chance in EACH game, so yes

29. Let’s Try an Example
A football team has a 40% chance of winning each of their games. Over the course of a 12-game season, what is the probability that they win exactly 7 games?
Binompdf(12, .4, 7) = .1009
What about winning 7 OR MORE games?

30. Let’s Try an Example
A football team has a 40% chance of winning each of their games. Over the course of a 12-game season, what is the probability that they win exactly 7 games?
What about winning 7 OR MORE games?
1- Binomcdf(12, .4, 6)= .1582

31. Now you try
A study showed that people cannot tell the difference between tap water and bottled water. A group of students place 3 cups in front of all of their teachers, one of which contains bottled water, and they ask the teachers to identify which one in a taste test. Teachers are not allowed to talk to each other. The experiment is going to test how many of the 31 teachers can guess/identify the bottled water correctly
Identify whether this is a binomial setting
Assuming that it is a binomial setting, what is the probability of exactly 16 teachers guessing correctly?
What is the probability that less than half of teachers guess correctly?
What is the probability that more than 18 teachers guess correctly?

32. Answers
YES
.0159
.973
.0013

33. Getting Credit on the AP Test
On multiple choice, not an issue
On free response, need to show a little bit of work
Unfortunately, just writing binompdf(31, 1/3, 16) doesn’t get full credit
Several ways to get credit:
Show the binomial probability formula with values plugged in. e.g. 𝑃 𝑋=16 = ( )
Write it in a sentence: “I used binompdf(31, 1/3, 16) on my calculator with n=31, p=1/3, and k=16)”
Define X clearly. “X is a binomial random variable where n=31 and p=1/3. I therefore use binompdf(31, 1/3, 16) on my calculator

35. Mean and Standard Deviation
Both formulas are on the AP formula sheet

36. Quick example So in our bottled water example N=31 P=1/3
So the mean would be (31)(1/3)=
Standard deviation would be: (31)( 1 3 )( 2 3 ) =2.625

37. Last Thing!
If you have a population containing p proportion of successes…
And you take an SRS of size n from the population
If the sample is no larger than 10% of the population, then…
The binomial distribution with n and p can be used to approximate the number of successes in the sample