1. Chapter 11
Probability

2. Chapter 11: Probability 11.1 Basic Concepts
11.2 Events Involving “Not” and “Or”
11.3 Conditional Probability and Events Involving “And”
11.4 Binomial Probability
11.5 Expected Value and Simulation

3. Section 11-4
Binomial Probability

4. Binomial Probability
Construct a simple binomial probability distribution.
Apply the binomial probability formula for an experiment involving Bernoulli trials.

5. Binomial Probability Distribution
The spinner below is spun twice and we are interested in the number of times a 2 is obtained (assume each sector is equally likely).
Think of outcome 2 as a “success” and outcomes 1 and 3 as “failures.” The sample space is 2 1 3
S = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2),
(2, 3), (3, 1), (3, 2), (3, 3)}.

6. Binomial Probability Distribution
When the outcomes of an experiment are divided into just two categories, success and failure, the associated probabilities are called “binomial.” Repeated trials of the experiment, where the probability of success remains constant throughout all repetitions, are also known as Bernoulli trials.

7. Binomial Probability Distribution
If x denotes the number of 2s occurring on each pair of spins, then x is an example of a random variable. In S, the number of 2s is 0 in four cases, 1 in four cases, and 2 in one case. Because the table on the next slide includes all the possible values of x and their probabilities, it is an example of a probability distribution. In this case, it is a binomial probability distribution.

8. Probability Distribution for the Number of 2s in Two Spinsx
P(x) 1 2

9. Binomial Probability Formula
In general, let
n = the number of repeated trials,
p = the probability of success on any given trial,
q = 1 – p = the probability of failure on any given trial,
and x = the number of successes that occur.
Note that p remains fixed throughout all n trials. This means that all trials are independent. In general, x successes can be assigned among n repeated trials in nCx different ways.

10. Binomial Probability Formula
When n independent repeated trials occur, where
p = probability of success and
q = probability of failure
with p and q (where q = 1 – p) remaining constant throughout all n trials, the probability of exactly x successes is given by

11. Example: Finding Probability in Coin Tossing
Find the probability of obtaining exactly three heads in five tosses of a fair coin.
Solution
This is a binomial experiment with n = 5, p = 1/2, q = 1/2, and x = 3.

12. Example: Finding Probability in Dice Rolling
Find the probability of obtaining exactly two 3’s in six rolls of a fair die.
Solution
This is a binomial experiment with n = 6, p = 1/6, q = 5/6, and x = 2.

13. Example: Finding Probability in Dice Rolling
Find the probability of obtaining less than two 3’s in six rolls of a fair die.
Solution
We have n = 6, p = 1/6, q = 5/6, and x < 2.

14. Example: Finding the Probability of Hits in Baseball
A baseball player has a well-established batting average of In the next series he will bat 10 times. Find the probability that he will get more than two hits.
Solution
In this case n = 10, p = 0.250, q = 0.750, and x > 2.

15. Example: Finding the Probability of Hits in Baseball
Solution (continued)