Probability Distributions: Binomial

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Probability Distributions Binomial Normal

• Ginger Holmes Rowell, PhD
• MSP Workshop
• June 2006

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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

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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

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Use the Table as a Guide
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X number of tails in 2 tosses
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Draw a graph representing the distribution of X
( of tails in 2 flips)
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Some Terms to Know

• Random Experiment
• Random Variable
• Discrete Random Variable
• Continuous Random Variable
• Probability Distribution

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Terms

• Random Experiment
• Examples

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Terms Continued

• Random Variable
• Examples

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Terms Continued

• Discrete Random Variable
• Example
• Continuous Random Variable
• Example

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Terms Continued

• The Probability Distribution of a random
  variable, X,
• Example

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X counts the number of tails in two flips of a
coin
Specify the random experiment the random
variable for this probability distribution.
Is the RV discrete or continuous?
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Properties of Discrete Probability Distributions
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Mean of a Discrete RV

• Mean value
• Example X counts the number of tails showing in
  two flips of a fair coin
• Mean

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Example Your Turn

• Example 12, parental involvement

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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

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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

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Binomial Example
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What is the Binomial Probability Distribution?

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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) ____

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Mean of a Binomial RV

• Example Test guessing
• In general mean
• Variance

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Using the TI-84

• To find P(Xa) for a binomial RV for an
  experiment with n trials and probability of
  success p
• Binompdf(n, p, a) P(Xa)
• Binomcdf(n, p, a) P(X lt a)

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Pascals Triangle Binomial Coefficients

• Handout
• Pascals Triangle Applet
• http//www.mathforum.org/dr.cgi/pascal.cgi?rows10
  

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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 lt 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?

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Airplane Example
Source NCTM Standards for Prob/Stat.
D\Standards\document\chapter6\data.htm
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Airplane Problem

• A Probability

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Homework

• Blood type problem
• Handout 22, 26, 37

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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

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Continuous Distributions

• Probability Density Function

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Example Normal Distribution

• Draw a picture
• Show Probabilities
• Show Empirical Rule

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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

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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?

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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

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Using your calculator

• Normalcdf ( a, b, mean, st dev)
• Use the calculator to solve problems on the
  previous page.

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Homework

• Handout s 12, 14, 15, 16, 24