1. The Binomial Probability Distribution
6.2
The Binomial Probability Distribution

2. Binomial Experiment A binomial experiment has the following structure
The first test is performed … the result is either a success or a failure
The second test is performed … the result is either a success or a failure. This result is independent of the first and the chance of success is the same
A third test is performed … the result is either a success or a failure. The result is independent of the first two and the chance of success is the same

3. Binomial Experiment
A card is drawn from a deck. A “success” is for that card to be a heart … a “failure” is for any other suit
The card is then put back into the deck
A second card is drawn from the deck with the same definition of success.
The second card is put back into the deck
We continue for 10 cards

4. Binomial Experiment
A binomial experiment is an experiment with the following characteristics
The experiment is performed a fixed number of times, each time called a trial
The trials are independent
Each trial has two possible outcomes, usually called a success and a failure
The probability of success is the same for every trial

5. Notation for Binomial Experiments
Notation used for binomial distributions
The number of trials is represented by n
The probability of a success is represented by p
The total number of successes in n trials is represented by X
Because there cannot be a negative number of successes, and because there cannot be more than n successes (out of n attempts)
0 ≤ X ≤ n

6. In our card drawing example (drawing a heart)
Each trial is the experiment of drawing one card
The experiment is performed 10 times, so n = 10
The trials are independent because the drawn card is put back into the deck
Each trial has two possible outcomes, a “success” of drawing a heart and a “failure” of drawing anything else
The probability of success is 0.25, the same for every trial, so p = 0.25
X, the number of successes, is between 0 and 10

7. Success?
The word “success” does not mean that this is a good outcome or that we want this to be the outcome
A “success” in our card drawing experiment is to draw a heart
If we are counting hearts, then this is the outcome that we are measuring
There is no good or bad meaning to “success”

8. Binomial Probability Formula
The general formula for the binomial probabilities is just this
For P(x), the probability of x successes, the probability is
The number of ways of choosing x out of n, times
The probability of x successes, times
The probability of n-x failures
This formula is
P(x) = nCx px (1 – p)n-x

9. Calculators
We do not need to use the formula, we will be using the calculator commands instead.
We will do more of these at the end using technology

10. Mean of a Binomial Distribution
We would like to find the mean (expected value) of a binomial distribution.
Example
There are 10 MC questions on a quiz
The probability of success is .20 on each one
Then the expected number of successes (correct guesses on the quiz) would be 10 • .20 = 2
The general formula: μX = n p

11. Variance of a Binomial Distribution
We would like to find the standard deviation and variance of a binomial distribution
This calculation is more difficult
The standard deviation is σX = 𝒏 𝒑 (𝟏 – 𝒑) and the variance is σX2 = n p (1 – p)

12. Example For our random guessing on a quiz problem Therefore
The mean is np = 10 • .2 = 2
The variance is np(1-p) = 10 • .2 • .8 = 1.6
The standard deviation is √1.6 = 1.26
Remember the empirical rule? A passing grade of 6 is 10 standard deviations from the mean …

13. Binomial Probability Distribution
With the formula for the binomial probabilities P(x), we can construct histograms for the binomial distribution
There are three different shapes for these histograms
When p < .5, the histogram is skewed right
When p = .5, the histogram is symmetric
When p > .5, the histogram is skewed left

14. Skewed Right For n = 10 and p = .2 (skewed right) Mean = 2
Standard deviation = .4

15. Skewed Left For n = 10 and p = .8 (skewed left) Mean = 8
Standard deviation = .4

16. Symmetric For n = 10 and p = .5 (symmetric) Mean = 5
Standard deviation = .5

17. Shape
Despite binomial distributions being skewed, the histograms appear more and more bell shaped as n gets larger
This will be important!

18. Calculators
BinomCdf and BinomPdf

19. Calculator Instructions
MENU 5:Probability, 5:Distributions, D: BinomialPDF
Use PDF if you are finding probability of a specific number of successes.
Enter n(# of trials), p(probability of success), x (# of successes desired)
If you leave x blank, you will get a list of all possible successes not just a certain #
MENU 5:Probability, 5:Distributions, E: BinomialCDF
Use CDF if you are finding the probability of a range of successes.
Enter n(# of trials), p(probability of success), then the lower and upper bounds of the range you are finding

20. Example Suppose 70% of all Americans have cable TV.
P(x) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Suppose 70% of all Americans have cable TV.
In a Random sample of 15 households, Construct a binomial probability distribution.

21. Example Suppose 70% of all Americans have cable TV.
In a Random sample of 15 households, what is the probability that exactly 10 households have cable?
Binompdf(n, p, x)
Binompdf(15, .7, 10)
Use pdf when exactly is used!!!
Use Calculator and: P(x = 10) =

22. Another Example Part 2 Suppose 70% of all Americans have cable TV.
In a Random sample of 15 households, what is the probability that at least 13 households have cable?
P( x 13) = P(13) or P(14) or P(15)

23. Another Example Part 3 Suppose 70% of all Americans have cable TV.
In a Random sample of 15 households, what is the probability that fewer than 13 households have cable?
P(x )= P(1)+P(2)….P(12)
Notice the 12, NOT 13!!

24. Example (#35)
According to flightstats.com, American Airlines flight 1247 from Orlando to Los Angeles is on time 65% of the time. Suppose fifteen flights are randomly selected, and the number of on-time flights is recorded.
Explain why this is a binomial experiment
Find the probability that exactly 10 flights are on time.
Find the probability that at least 10 flights are on time.
Find the probability that fewer that 10 flights are on time.
Find the probability that between 7 and 10 flights, inclusive, are on time.

25. Example (#49)
According to the Uniform Crime Report, 2006, nationwide, 61% of murders committed in 2006 were cleared by arrest or exceptional means.
For 250 randomly selected murders committed in 2006, compute the mean and standard deviation of the random variable x, the number of murders cleared by arrest or exceptional means.
Interpret the mean
Of the 250 randomly selected murders, find the interval that would be considered “unusual” for the number of murders that was cleared.