Post on 29-Dec-2015
Probability Distributions: Binomial & Normal
Ginger Holmes Rowell, PhD
MSP Workshop
June 2006
Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
Start with an Example
Flip two fair coins twice List the sample space:
Define X to be the number of Tails showing in two flips.
List the possible values of X Find the probabilities of each value of X
Use the Table as a Guide
x Probability of getting “x”
0
1
2
X = number of tails in 2 tosses
x Probability of getting “x”
0 P(X=0) = P(HH) = .25
1 P(X=1) = P(HT or TH) = .5
2 P(X=0) = P(HH) = .25
Draw a graph representing the distribution of X (# of tails in 2 flips)
Some Terms to Know
Random Experiment
Random Variable
Discrete Random Variable Continuous Random Variable
Probability Distribution
Terms
Random Experiment:
Examples:
Terms Continued
Random Variable:
Examples
Terms Continued
Discrete Random Variable
Example
Continuous Random Variable
Example
Terms Continued
The Probability Distribution of a random variable, X,
Example:
X counts the number of tails in two flips of a coin
x Probability of getting “x”
0 P(X=0) = .25
1 P(X=1) = .50
2 P(X=2) = .25
Specify the random experiment & the random variable for this probability distribution.
Is the RV discrete or continuous?
Properties of Discrete Probability Distributions
Mean of a Discrete RV
Mean value =
Example: X counts the number of tails showing in two flips of a fair coin Mean =
Example: Your Turn
Example # 12, parental involvement
Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
Binomial Distribution
If X counts the number of successes in a binomial experiment, then X is said to be a binomial RV. A binomial experiment is a random experiment that satisfies the following
Binomial Example
What is the Binomial Probability Distribution?
Binomial Distribution
Let X count the number of successes in a binomial experiment which has n trials and the probability of success on any one trial is represented by p, then
Check for the last example: P(X = 2) = ____
Mean of a Binomial RV
Example: Test guessing
In general: mean = Variance =
Using the TI-84
To find P(X=a) for a binomial RV for an experiment with n trials and probability of success p
Binompdf(n, p, a) = P(X=a)
Binomcdf(n, p, a) = P(X <= a)
Pascal’s Triangle & Binomial Coefficients
Handout
Pascal’s Triangle Applet http://www.mathforum.org/dr.cgi/pascal.cgi
?rows=10
Using Tree Diagrams for finding Probabilities of Complex Events
For a one-clip paper airplane, which was flight-tested with the chance of throwing a dud (flies < 21 feet) being equal to 45%. What is the probability that exactly one of
the next two throws will be a dud and the other will be a success?
Airplane Example
Source: NCTM Standards for Prob/Stat. D:\Standards\document\chapter6\data.htm
Airplane Problem
A: Probability =
Homework
Blood type problem Handout # 22, 26, 37
Overview
Some Important Concepts/Definitions Associated with Probability Distributions
Discrete Distribution Example: Binomial Distribution More practice with counting and complex
probabilities Continuous Distribution Example:
Normal Distribution
Continuous Distributions
Probability Density Function
Example: Normal Distribution
Draw a picture Show Probabilities Show Empirical Rule
What is Represented by a Normal Distribution?
Yes or No Birth weight of babies born at 36 weeks Time spent waiting in line for a roller
coaster on Sat afternoon? Length of phone calls for a give person IQ scores for 7th graders SAT scores of college freshman
Penny Ages
Collect pennies with those at your table. Draw a histogram of the penny ages Describe the basic shape Do the data that you collected follow the
empirical rule?
Penny Ages Continued
Based on your data, what is the probability that a randomly selected penny is is between 5 & 10 years old? Is at least 5 years old? Is at most 5 years old? Is exactly 5 years old? Find average penny age & standard
deviation of penny age
Using your calculator
Normalcdf ( a, b, mean, st dev)
Use the calculator to solve problems on the previous page.
Homework
Handout #’s 12, 14, 15, 16, 24