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





List the sample space:
S = { HH, HT, TH, TT }
Define X to be the number of Tails showing
in two flips.
List the possible values of X = {0, 1, 2}
Find the probabilities of each value of X
Use the Table as a Guide
x
0
1
2
Probability of
getting “x”
Use the Table as a Guide
x = # of
Probability of getting “x”
tails
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: an experiment
that can be repeated under the same
conditions and you do not outcome of
the experiment in advance
 Examples:



Flip a coin
Roll a pair of dice
Survival times of persons with a given
disease
Terms Continued
 Random Variable: a numerical
representation of the outcome of a
random experiment
 Examples



Flip a coin twice: X counts the number of
tails showing
Rolling two dice: X represents the sum of
the two faces showing
Survival times: number of months people
live once diagnosed
Terms Continued
 Discrete Random Variable

A RV that has a finite (or countable)
number of possible outcomes
 Example

Sum of faces showing when roll two dice
 Continuous Random Variable

A RV that has an infinite (uncountable)
number of possible outcomes
 Example

Birth weight, time spent doing homework
Terms Continued
 The Probability Distribution of a random
variable, X, is a table, chart, graph or
formula which specifies the probabilities
for all possible values in the sample
space (i.e. for all possible values of the
random variable).
 Example: Let’s go back to our original
example of flipping a fair coin twice.
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
 The sum of the probabilities of all items
equals one
 Each individual item’s probability is
between 0 and 1 (inclusive)
 Example: Binomial Distribution
 Probability Histogram

Horizontal axis shows values of the RV &
vertical axis represents the probability that
the corresponding value of the RV occurs.
Mean of a Discrete RV
 Mean value = sum of ( value of x *
probability the given value occurs )
 Example: X counts the number of tails
showing in two flips of a fair coin

Mean = sum of [ x*P(X=x) ] over all x’s
= 0*.25 + 1*.5 + 2 *.25
= 1 tail showing
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




Each trial ends in a success or failure
P(Success) is the same for every trial
There are a finite number of trials
Trials are independent.
Binomial Example
 Flip a fair coin twice
 Rolling doubles on two dice
 Shooting 10 free throws
…
What is the Binomial
Probability Distribution?
 Example: Flip a fair coin twice

We already answered this problem
 Example: Test Guessing Handout
 Generate General Formula
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
n a
na
P( X  a)    p (1  p)
for x  0, 1, 2,...n
a
Check for the last example: P(X = 2) = ____
Mean of a Binomial RV
 Example: Test guessing
 In general: mean = n*p
 Variance = n*p*(1-p)
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 = 198/400, or .495, since
each of the two possibilities—"dud first,
then success" and "success first, then
dud"—has a probability of 99/400.
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

An equation used to compute probabilities
of continuous random variables that satisfy
the following
 Area
under the graph of the equation over all
the possible values of the RV must equal one
 The graph of the equation must be >= zero
for all possible values of the RV
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 coster
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