1. LESSON 9: BINOMIAL DISTRIBUTION
Outline
The context
The properties
Notation
Formula
Use of table
Use of Excel
Mean and variance

2. BINOMIAL DISTRIBUTION THE CONTEXT
An important property of the binomial distribution:
An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure.
Example: Suppose that a production lot contains 100 items. The producer and a buyer agree that if at most 2 out of a sample of 10 items are defective, then all the remaining 90 items in the production lot will be purchased without further testing. Note that each item can be defective or non defective which are two mutually exclusive outcomes of testing. Given the probability that an item is defective, what is the probability that the 90 items will be purchased without further testing?

3. BINOMIAL DISTRIBUTION THE CONTEXT
Trial Two Mut. Excl. and exhaustive outcomes
Flip a coin Head / Tail
Apply for a job Get the job / not get the job
Answer a Multiple Correct / Incorrect
choice question

4. BINOMIAL DISTRIBUTION THE PROPERTIES
The binomial distribution has the following properties:
1. The experiment consists of a finite number of trials. The number of trials is denoted by n.
2. An outcome of an experiment is classified into one of two mutually exclusive categories - success or failure.
3. The probability of success stays the same for each trial. The probability of success is denoted by π.
4. The trials are independent.

5. BINOMIAL DISTRIBUTION THE NOTATION
n : the number of trials
r : the number of observed successes
π : the probability of success on each trial
Note:
n-r : the number of observed failures
1- π : the probability of failure on each trial

6. BINOMIAL DISTRIBUTION THE PROBABILITY DISTRIBUTION
The binomial probability distribution gives the probability of getting exactly r successes out of a total of n trials.
The probability of getting exactly r successes out of a total of n trials is as follows:
Note: In the above gives the number of different ways of choosing r objects out of a total of n objects

7. BINOMIAL DISTRIBUTION THE PROBABILITY DISTRIBUTION
Example 1: If you toss a fair coin twice, what is the probability of getting one head and one tail? Use the binomial probability distribution formula.

8. BINOMIAL DISTRIBUTION THE PROBABILITY DISTRIBUTION
Example 2: Redo Example 1 with a probability tree and verify if the probability tree gives the same answer.

9. BINOMIAL DISTRIBUTION THE CUMULATIVE PROBABILITY
The cumulative probability gives the probability of getting at most r successes out of a total of n trials.
The probability of getting at most r successes out of a total of n trials is as follows:
Note: An uppercase B(r) is used to distinguish the cumulative probability distribution function from the probability mass function b(r)

10. BINOMIAL DISTRIBUTION THE CUMULATIVE PROBABILITY
Example 3: If you toss a fair coin three times, what is the probability of getting at most one head (at least two tails)?

11. BINOMIAL DISTRIBUTION THE CUMULATIVE PROBABILITY
Example 4: Redo Example 3 with a probability tree and verify if the probability tree gives the same answer.

12. BINOMIAL DISTRIBUTION NECESSITY OF A TABLE OR SOFTWARE
Example 5 (do not solve): If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Do not solve this problem, but discuss the computation required by the binomial probability distribution formula.

13. BINOMIAL DISTRIBUTION USE OF TABLE
Table A, Appendix A, pp gives the probability of getting at most r successes out of a total of n trials, for probability of success in each trial π.
The table can be used to find the probability of
exactly r successes:
at least r successes:
successes between a and b:

14. BINOMIAL DISTRIBUTION USE OF TABLE
Example 6: Find the following using Table A:
Example 7: Find the following using above values

15. BINOMIAL DISTRIBUTION USE OF TABLE
Example 8: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using the Table A.

16. BINOMIAL DISTRIBUTION USE OF EXCEL
The Excel function BINOMDIST gives and
It takes four arguments. The first 3 arguments are r,n,π
The last one is TRUE for and FALSE for
Example 9: If you toss a fair coin 50 times, what is the probability of getting at most 20 heads (at least 30 tails)? Solve this problem using Excel. Verify if Excel gives the same answer as it is given by Table A in Example 8.
Answer:
=BINOMDIST(20,50,0.5,TRUE)

17. BINOMIAL DISTRIBUTION MEAN AND VARIANCE
The expected value and variance for the number of successes R may be computed as follows:
E(R) is the mean or expected value of R
Var(X) is the variance of R
n is the number of trials
π is the probability of success on each trial
The probability of failure on each trial = 1- π

18. BINOMIAL DISTRIBUTION MEAN AND VARIANCE
Example 10: Let R be a random variable that gives number of heads when a fair coin is tossed 4 times. Compute E(R) and Var(R).

19. READING AND EXERCISES Lesson 9 Reading: Section 7-3, pp. 204-215
7-22, 7-24, 7-30