Transcript Probability Distributions: Binomial & Normal

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
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List the sample space:
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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
x
0
1
2
Probability of
getting “x”
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
0
Probability of getting “x”
1
P(X=1) = .50
2
P(X=2) = .25
P(X=0) = .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
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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
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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