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Ch5: Discrete Probability Distributions
Section 5-1: Probability Distribution
Recall:
A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g. X , Y , Z ) are used to represent variables. A random variable is a variable whose values are determined by chance. Discrete variables are countable.
Example: Roll a die and let X represent the outcome so X = {1,2,3,4,5,6}
Discrete probability distribution - the values a random
variable can assume and the corresponding probabilities of
the values.
The probabilities may be determined theoretically or by
observation.
They can be displayed by a graph or a table.
How does this connect to our frequency distributions,
tables and graphs from Chapter 2?
Graph the probability distribution above.
0.0 0.5 1.0 1.5 2.0 2.5 3.
Number of Girls
Probability
Example: The World Series played by Major League Baseball is a 4 to 7 game series won by the team winning four games. The data shown consists of the number of games played in the World Series from 1965 through 2005. The number of games played is represented by the variable X.
Two Requirements for a Probability Distribution
1. The sum of the probabilities of all the outcomes in the
sample space must be 1; that is
(^) P ( X ) 1.
2. The probability of each outcome in the sample space
must be between or equal to 0 and 1; that is
0 P ( X ) 1.
These are good checks for you to use after you have computed a discrete probability distribution! The “sums to 1” check will often find a calculation error!
Example: Determine whether each distribution is a probability distribution. Explain.
Example : Find the mean number of girls in a family with two children using the probability distribution below.
X P X
X
Example : Find the mean number of trips lasting five nights or longer that American adults take per year using the probability distribution below.
(^) X P ( X )
Example : Calculate the variance and standard deviation for the number of girls in the previous example:
2 2 2
2 2
2 2 2
[ ( )]
[ ( )] 1
1 1 1 3 1 0 1 2 1 1 4 2 4 2 2
X P X
X P X
(^) (^)
2 1
2
Example : Calculate the variance and standard deviation for the number of trips five nights or more in the previous example.
2 (^) [ X^2 P ( X )]^2
Example: Suppose one thousand tickets are sold at $10 each to win a used car valued at $5,000. What is the expected value of the gain if a person purchases one ticket?
The person will either win or lose. If they win which will happen with probability 1/1000, they have gained $5000-$10. If they lose, they have lost $10.
Gain, X Probability, P ( X ) $4990 1/
- $10 999/
1 999 $4990 ( $10) $ 1000 1000
E X
Example: Suppose one thousand tickets are sold at $1 each for 3 prizes of $150, $100, and $50. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if a person purchases 3 tickets?
Gain, X Probability, P ( X )
E(X) =
Section 5-3: The Binomial Distribution
A binomial experiment is a probability experiment that satisfies the following four requirements:
- Each of the n trials has two possible outcomes or can be reduced to two outcomes: “success” and “failure”. The outcome of interest is called a success and the other outcome is called a failure.
- The outcomes of each trial must be independent of each other.
- There must be a fixed number of trials.
- Each trial has the same probability of success, denoted by p.
The acronym BINS may help you remember the conditions:
B – Binary outcomes
I – Independent outcomes
N – number of trials is fixed
S – same probability of success
Examples:
