Discrete Probability Distributions

Dr. Rick Jerz

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GOALS

• Define the terms probability distribution and random variable.

• Distinguish between discrete and continuous probability distributions.

• Calculate the mean, variance, and standard deviation of a discrete probability distribution.

• Describe the characteristics of and compute probabilities using the binomial probability distribution.

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What is a Probability Distribution?

Experiment: Toss a coin three times. Observe the number of heads. The possible results are: zero heads, one head, twoheads, and three heads. What is the probability distribution for the number of heads?

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A Distribution

10

9

3 32

012345678910

1 2 3 4 5 6

Coun

t

# in Bag

# of Green M&M's in Bag

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A Probability Distribution

5.6%0.0%

50.0%

16.7% 16.7%11.1%

0.0%0%

10%

20%

30%

40%

50%

60%

1 2 3 4 5 6 7

Perc

ent

# in Bag

Probability of Green M&M's in Bag

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Probability for Die Toss

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Tossing 2 Dice

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Probability Distribution, # Heads, 3 Coin Tosses

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Characteristics of All Probability Distributions

• The probability of a particular outcome is between 0 and 1 inclusive

• The outcomes are mutually exclusive events• The list is exhaustive, the sum of the

probabilities of the outcomes is equal to 1

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Random Variables

• Random variable - a quantity resulting from an experiment that, by chance, can assume different values.

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Types of Random Variables

• Discrete Random Variable can assume only certain clearly separated values. It is usually the result of counting something.

• Continuous Random Variable can assume an infinite number of values within a given range. It is usually the result of some type of measurement.

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Discrete Random Variables Examples

• The number of students in a class• The number of children in a family• The number of cars entering a carwash

in an hour• Number of home mortgages approved by

Coastal Federal Bank last week

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Continuous Random Variables Examples

• The distance students travel to class• The time it takes an executive to drive to

work• The length of an afternoon nap• The length of time of a particular phone call

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Characteristics of a DiscreteDistribution

• The main features of a discrete probability distribution are:• The sum of the probabilities of the various

outcomes is 1.00• The probability of a particular outcome is

between 0 and 1.00• The outcomes are mutually exclusive

and• The item being measured can assume only

certain separated (counted) values.

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The Mean of a Probability Distribution

• The mean is a typical value used to represent the central location of a probability distribution

• The mean of a probability distribution is also referred to as its expected value

[ ( )]xP xµ =å𝜇 = # 𝑥𝑃(𝑥)

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Similar to “Weighted Mean”

• The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

1 1 2 2

1 2

n nw

n

w X w X w XXw w w+ + +

=+ + +

!

!

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The Variance and StandardDeviation

• Measures the amount of spread in a distribution

• The computational steps are:1. Subtract the mean from each value, and square

this difference2. Multiply each squared difference by its

probability3. Sum the resulting products to arrive at the

variance

• The standard deviation is found by taking the square root of the variance

2 2[( ) ( )]x u P xs = -å𝜎! =# 𝑥 − 𝜇 !𝑃(𝑥)

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Mean, Variance, and StandardDeviation Example

• John Ragsdale sells new cars for Pelican Ford. John usually sells the largest number of cars on Saturday. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.

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Mean of a Probability Distribution Example

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Variance and StandardDeviation Example

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Binomial Probability Distribution (Discrete)

Characteristics:• There are only two possible outcomes on a

particular trial of an experiment• The outcomes are mutually exclusive • The random variable is the result of counts• Each trial is independent of any other trial• Examples:• Yes or no• True or false• On or off• Correct or incorrect

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Tree Diagrams

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Binomial Tree

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Binomial Probability Formula

Where• C denotes a combination• n is the number of trials• x is the random variable defined as the

number of successes• π is the probability of a success on each trial

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Example with M&M’s

Brown M&M (Binomial)

0.00

0.20

0.40

0.60

0.80

1.00

0 >0

2 Conditions

Prob

abili

ty

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Question:

• What is the probability that 2 out of 4 bags have brown M&M’s?• N=4, x=2, π = .86

• How about 3 out of 4 bags having brown M&M’s?• N=4, x=3, π = .86

• How about between 2 and 3 out of 4 bags having brown M&M’s?• N=4, x=3?, π = .86

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Example: Binomial Probability

• There are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport, Pennsylvania. Suppose the probability that any flight arrives late is .20.

• What is the probability that none of the flights are late today?

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Binomial DistributionMean and Variance

• Mean of a binomial distribution

• Variance of a binomial distribution

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Binomial Dist. – Mean and Variance: Example

• For the example regarding the number of late flights, recall that p =.20 and n = 5.

• What is the average number of late flights?

• What is the variance of the number of late flights?

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Binomial Dist. - Mean and Variance: Another Solution

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Binomial Distribution - Table• Five percent of the worm gears produced by an

automatic, high-speed Carter-Bell milling machine are defective. What is the probability that out of six gears selected at random none will be defective? Exactly one? Exactly two? Exactly three? Exactly four? Exactly five? Exactly six out of six?

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Binomial – Shapes for Varying p(n constant)

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Binomial – Shapes for Varying n with (p constant)

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Cumulative Binomial Probability Distributions

• A study in June 2003 by the Illinois Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?

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Binomial Probability - Excel

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Cumulative Binomial Probability Dist. In Excel

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