1. The Binomial Distribution
Chapter 7 Section 3

2. When a probability problem can be reduced to two outcomes or only has two outcomes, it is called a binomial experiment. It must also meet the following requirements: a.) must be a fixed number of trials b.) outcomes of trials must be independent c.) probability for a success must remain the same for each trial.
Binomial Experiment

3. Binomial Distribution
The outcomes of a binomial experiment and the corresponding outcomes are called a binomial distribution. Notation for Binomial Distribution: P(S) symbol for the probability of success P(F) symbol for the probability of failure p numerical probability of success q numerical probability of failure P(S) = p P(F) = 1 – p = q n number of trials X number of successes in n trials
Binomial Distribution

4. Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is:
Binomial Probability Formula

5. A coin is tossed 3 times. Find the probability of getting exactly two heads. This is a binomial experiment because: 1.) Fixed number of trials (n = 3) 2.) There are only tow outcomes for each trial (heads or tails) 3.) The outcomes are independent 4.) The probability of success (heads) is ½ in each case.
Example #1

6. We can solve this problem simply by looking at the sample space:
Example #1 Cont.

7. In this problem: n = 3, X = 2, p = ½ and q = ½
Example #1 Cont.

8. A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. n = X = p = q =
Example #2

9. A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them have part time jobs. (To find the probability of at least, we must find the probability of 3, 4, and 5 and then add them together)
Example #3

10. Public Opinion reported that 5% of Americans are afraid of being alone in a house at night. If a random sample of 20 Americans is selected, find the probability there are exactly 5 people in the sample who are afraid of being alone at night.
Example #4

11. Twenty-five percent of the customers entering a grocery store between 5 PM and PM use an express checkout. Consider 5 randomly selected customers and let x denote the number among the five who use the express checkout.
What is p(2)?
What is (P<=1)
What is P(>=2)
Example #5

12. Mean, Variance, and Standard Deviation for the Binomial Distribution
Mean: Variance: Standard Deviation:
Mean, Variance, and Standard Deviation for the Binomial Distribution

13. A coin is tossed 4 times. Find the mean, variance, and standard deviation. A die is rolled 480 times. Find the mean, variance and standard deviation.
Example #6 and #7

14. Geometric Distribution
Chapter 7 Section 5

15. Geometric Random Variable
A geometric random variable is defined as the number of trials (x) until the first success is observed. The probability distribution of x is called the geometric probability distribution.
Geometric Random Variable

16. Geometric Probability Distribution Equation

17. Cumulative Geometric Distribution Equation
And
Cumulative Geometric Distribution Equation

18. Expected value

19. You have left your lights on and your car battery has died
You have left your lights on and your car battery has died. Assume that 20% of students who drive to school carry jumper cables. What is the probability that the first student you ask has jumper cables?
What is the probability that 3 or fewer students must be stopped?
What is the expected number before someone will have jumper cables?
Example #1

20. One out of seven scratch off lottery tickets has a prize
One out of seven scratch off lottery tickets has a prize. You keep buying tickets until you win a prize, then you stop.
Find the probability that you buy 4 tickets.
Find the probability that you buy 3 or fewer tickets
Find the probability that you buy more than four tickets
What is the expected number of tickets bought?
Example #2