Probability Concepts

Dr. Rick Jerz

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Goals

• Probability Concepts• Define probability, experiment, event, and

outcome• Graph event probability using probability

distributions• Define the terms “conditional probability” and

“joint probability”• Calculate probabilities using the rules of addition

and rules of multiplication• Graph multiple events using a tree diagram

• Describe the classical, empirical, and subjective approaches to probability

• Calculate the number of arrangements, permutations, and combinations

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Topic Overview

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Examples of Business Decisions

• If we revise our product, will customers purchase it?

• Should we guarantee our company’s tires to last 40,000 miles?

• Should I hire a new employee?• If I am dealt five cards, will I get a royal flush?• Will the next die throw produce a 4?• Will the toss of a coin be heads or tails?• Will Hillary Clinton be the next president?• Can a student guess on a 10-question

multiple choice test and earn an 85%?

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Helpful Definitions

• An experiment is the observation of some activity or the act of taking some measurement

• Outcomes are the possible results of an experiment

• An event is the collection of one or more outcomes of an experiment

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Experiments, Outcomes, and Events

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What is probability?

• Probability is a measure of the likelihood that an event in the future will happen. It can only assume a value between 0 and 1, or 0 and 100%

• A value near zero means the event is not likely to happen. A value near 1 means it is likely.

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M&M Experiment

• Experiment: pick an M&M from a bag

• Outcomes: red, blue, brown, green, orange, or yellow

• Events of interest:• A red M&M• A red or orange M&M• A primary color M&M• A green M&M

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Probability Examples

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Chart Types for M&M’s“Probability Distribution”

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Frequency Diagram16%

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Pie Chart

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Relative Frequency Diagram

Probability Distribution

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Probability Distribution Graphs

• Experiment: Pick an M&M from the bag

• Outcomes: Blue, brown, green, orange, yellow, red

• Probability of event:• “Red” = 26%• “Red or orange” = 42%• “Primary color” = 63%• “Red and orange” = 0%

16% 16%

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Relative Frequency Diagram

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A More Difficult Probability Distribution

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Diagraming the Experiment

• Another event of interest:

“not red”

• = 16+16+5+16+21

• This can be a lot of work!

16% 16%

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Relative Frequency Diagram

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Some Probability Rules

• Complement• Addition• Joint probability? (mutually exclusive events)

• Multiplication• Conditional?

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The Complement Rule

• The complement rule is used to determine the probability of an event “not occurring” by subtracting the probability of the event occurring from 1.

P(A) + P(~A) = 1or P(A) = 1 - P(~A)

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Computing Event ProbabilitiesRule of Addition

Rules of Addition• Special Rule of Addition - If two

events A and B are mutually exclusive, the probability of one ORthe other events occurring equals the sum of their probabilities.

P(A or B) = P(A) + P(B)

• The General Rule of Addition - If A and B are two events that are not mutually exclusive, then subtract the joint probability, P(A and B), using the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

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Addition Rule - Example

• What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart?

• P(A or B) = P(A) + P(B) - P(A and B)• = 4/52 + 13/52 - 1/52• = 16/52, or .3077

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Joint ProbabilityVenn Diagram

• JOINT PROBABILITY: A probability that measures the likelihood two or more events will happen concurrently. P(A and B)

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Rule of Multiplications

• Used for independent events

• Independence means that they are not connected, e.g., not at the same time

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Probability of Multiple Events

• The special rule of multiplication requires that two events A and B are independent.

• This rule is written: • P(A and B) = P(A)P(B)

• The general rule of multiplication is used when the probability of event B is influenced, or conditioned, by the occurrence of event A.

• This rule is written: • P(A and B) = P(A)P(B|A)

(the conditional probability of B)

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Example: General Multiplication Rule

• A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.

• What is the likelihood both shirts selected are white?

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General Multiplication Rule-Example

• The event that the first shirt selected is white is W1 and the second shirt is W2

• The probability is P(W1) = 9/12 • P(W2) is P(W2 | W1) = 8/11• To determine the probability of 2 white shirts

being selected we use formula: P(A and B) = P(A) P(B|A)

• P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) = 0.55

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Independent EventsConditional Probability

• Questions, with or without replacement

red, then another red

red, then an orange

three reds on three tries

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Relative Frequency Diagram

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Tree Diagrams for Independent Events

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The Three Ways ofAssessing Probability

• Subjective: A guess, based upon intuition or judgment

• Empirical: Based upon relative frequency of sampled data (such as our M&M data)

• Classical: Based upon counting all possibilities that are equally likely within a population (such as rolling dice)

• Classical and empirical are “objective” (i.e., measured or calculated)

• Business Statistics Goal: Objective

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Subjective Probability

• If there is little or no past experience or information (data) on which to base a probability, it may be arrived at subjectively.

• Examples of subjective probability are:• Estimating the likelihood the New England

Patriots will play in the Super Bowl next year.• Estimating the likelihood that a new competitor

will enter the marketplace.• Estimating the likelihood the U.S. budget deficit

will be reduced by half in the next 10 years.

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Empirical Probability

• Don’t have all the population data, we collect sample data

• The probability of an event happening is the percent or fraction of the time similar events happened in the past

• Calculated by dividing theoccurrences of the event of interest by the total samplesize

• Bigger samples are better(more accurate). “Law of large numbers”

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Classical Probability

• We can count all possible outcomes of the experiment

• We can count all possible occurrences of the event (favorable outcomes) that we are interested in

• The calculated probability is the division of event outcome by all possible outcomes

• Challenge: can you count these “outcomes?”28

Summary of Types of Probability

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M&M Experiment:Empirical or Classical?

It depends…

• One bag as the entire (our class) population – classical

• Treating the data as a sample - empirical

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Calculating Probabilities: What is the Question?

• A king• A heart• A king or a heart• A king and a heart• A heart, if the king is not replaced• A king, then a heart (without replacement)• A king, then a heart (with replacement)• Many “not” questions, i.e., not a king

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Classical Probability

• Consider an experiment of rolling a six-sided die. What is the probability of the event “an even number of spots appear face up”?

• The possibleoutcomes are:

• There are three “favorable” outcomes (a two, a four, and a six) in the collection of six equally likely possible outcomes.

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“Randomness” Still Exists!• Suppose we toss a fair coin. The result of each toss

is either a head or a tail. If we toss the coin a great number of times, the probability of the outcome of heads will approach .5. The following table reports the results of an experiment of flipping a fair coin 1, 10, 50, 100, 500, 1,000 and 10,000 times and then computing the relative frequency of heads

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Counting all Outcomes

• For classical probability, we need to count the total number of outcomes and the events of interest

• Ways of counting• Arrangements• Permutations• Combinations

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Arrangements:Multiplication Formula

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Arrangement Example

• An automobile dealer wants to advertise that for $29,999 you can buy a convertible, a two-door sedan, or a four-door model with your choice of either wire wheel covers or solid wheel covers. How many different arrangements of models and wheel covers can the dealer offer?

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Another Example: Dell

• Website for laptops• How many different models does Dell have

for its Inspiron laptop?

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Another Example: Roll Two Dice

• What are all the outcomes?

• Solution: 6 possibilities for the first die, and 6 for the second, therefore 6x6 or 36

• Rolling a 5? (4,1 1,4 3,2 2,3) 4/36• Rolling a 7? (6,1 1,6 5,2 2,5 3,4 4,3) 6/36

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Results

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Painting a Room

• Our room has four walls, and we have 4 different colors of paint, each wall is a different color

• How many ways can we paint the room?• Blue, red, green, yellow

• 4*3*2*1• This is 4!

• Thus 4! = 4*3*2*1 = 2440

Painting a Room with Six Colors

• How would this differ if we had 6 colors of paint, and 4 walls?

• 6x5x4x3 = 360

• This is not quite 6!• We stop short

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Counting - Permutation

A permutation is the number of ways to choose r objects from a group n possible objects where the order of arrangements is important (abc and cba are different).

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Counting Arrangements for Subgroups w/o Replacement

• How many ways can we pick 4 colors from a total of 6 colors?

• Permutations (with order)• Combinations (without order)

• Note: # of permutations ≥ # combinations

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Counting - Combination

• A combination is the number of ways to choose r objects from a group of n objects without regard to order (abc and cba are not different).

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Example: Combination

There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible?

792)!512(!5

!12512 =

-=C

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Example: Permutation

Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability.

040,95)!512(!12

512 =-

=P

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Topic Overview

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Using Excel to Count

• Arrangements• Combinations• Permutations

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