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