1. The Binomial Probability Distribution
Section 6.2

2. Objectives
Determine whether a probability experiment is a binomial experiment
Compute probabilities of binomial experiments
Compute the mean and standard deviation of a binomial random variable
Construct binomial probability histograms

3. Characteristics of Binomial Probability Experiments
Binomial Probability Distribution:
Discrete
Describes probabilities for experiments in which there are only 2 mutually exclusive (disjoint) outcomes
Examples: yes/no, heads/tails,

4. Criteria for Binomial Probability Experiments
An experiment is said to be a binomial experiment if:
Performed a fixed number of times
Each time is called a trial
The trials are independent
The outcome of one trial will not affect the probability of the outcome of any of the other trials
For each trial, there are only 2 mutually exclusive (disjoint) outcomes: success or failure
The probability of success is the same for each trial of the experiment

5. Notation Used with the Binomial Probability Distribution
As always, a capital letter, X, is the notation for a random variable
There are always n independent trials of the experiment
Let p denote the probability of success, so that 1 – p is the probability of failure
Let x denote the number of successes in n independent trials of the experiment.
So 0 < x < n

6. Examples Examples of binomial experiments
Tossing a coin 30 times to see how many heads occur.
Asking 100 people if they watch CNN news.
Rolling a die to see if a 2 appears.
Examples which aren't binomial experiments
Rolling a die until a 4 appears (not a fixed number of trials)
Asking 50 people how old they are (not two outcomes)
Drawing 5 cards from a deck for a poker hand (done without replacement, so not independent)

7. Compute Probabilities of Binomial Experiments

8. Compute Probabilities of Binomial Experiments

9. Binomial Probability Distribution Function

10. Using the Binomial Probability Distribution Function
A coin is tossed 10 times. What is the probability that exactly 6 heads will occur?

11. Using the Binomial Probability Distribution Function
A coin is tossed 10 times. What is the probability that at least 5 heads will occur?

12. Using the Binomial Probability Distribution Function
A coin is tossed 10 times. What is the probability that at fewer than 3 heads will occur?

13. Using the Binomial Probability Distribution Function
A coin is tossed 10 times. What is the probability that heads will occur?

14. Assignment
Pg 340: 1, 3, 4, 5, 6, 7-16, 17, 20, 23, 25, 28

15. Computing Probabilities Using the Binomial Table
A coin is tossed 10 times. What is the probability that exactly 6 heads will occur?

16. Computing Using the TI-83/84

17. Computing Using the TI-83/84

18. Computing Using the TI-83/84

19. Compute the Mean and Standard Deviation of a Binomial Random Variable
These formulas are very simple and therefore there are no commands on the TI-83/84 calculator to replicate this.

20. Example
According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation number of car-owning households that will have three or more cars.

21. Construct Binomial Probability Histograms
Construct a binomial probability histogram with n=10 and p=0.2. What is the shape of this histogram? x
P(x) 1 2 3 4 5 6 7 8 9 10

22. Examples
(a) Construct a binomial probability histogram with n = 8 and p = 0.15.
(b) Construct a binomial probability histogram with n = 8 and p = 0. 5.
(c) Construct a binomial probability histogram with n = 8 and p = 0.85.

23. n = 8 and p = 0.15

24. n = 8 and p = 0. 5

25. n = 8 and p = 0.85

26. n = 50 and p = 0.8

27. n = 70 and p = 0.8

28. Number of Trials Affects Shape
For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped.
As a general rule of thumb, if np(1 – p) > 10, then the probability distribution will be approximately bell-shaped.

29. EXAMPLE. Using the Mean, Standard Deviation and
EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment
According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual ?
The result is unusual since 162 > 159.1

30. Assignment
Pg : 29, 30, 34, 36, 42, 43, 45, 47, 54