Introduc)on to Probability Module 2-1

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Learning Objec)ves

Roles of probability in decision making

Develop probability as a measure of uncertainty oAssign probabili)es

Basic rules of probabili)es

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What is probability?

Probability provides a numerical measure of the likelihood of an event occurring

Lies between 0 and 1 (inclusive) In an experiment, the sum of the probabili)es for all possible

outcomes is 1.

Increasing likelihood

0 0%

1 100%

0.5 50%

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Why/when do we use probability?

A measure of likelihood/uncertainty that can be used to improve decisions

Examples of decisions oChoose inventory levels for a product that has random demand

oDetermine how much to charge for dierent types of insurance policies

oPerform a cost-benet analysis of dierent alterna)ves

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Probability Deni)ons: What is an experiment, a sample point, and

sample space?

Experiment

Toss a coin Inspec)on a part Conduct a sales call Roll a die Play a football game

Possible outcomes (sample points)

Head, tail Defec)ve, non-defec)ve Purchase, no purchase 1, 2, 3, 4, 5, 6 Win, lose, )e

Sample Space: the collection of all possible outcomes

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Probability Deni)ons (Con)nued): What is an Event?

An event is a collec)on of sample points, i.e., a subset of the sample space.

Example: o Experiment: Roll a fair 6-sided die o Sample space: S = { 1, 2, 3, 4, 5, 6} o Let Event A = Ge`ng an even number when rolling

a die o A = {2, 4, 6} o P(A) = 3/6 = 0.5

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Note: The probability of any event is equal to the sum of the probabili)es of the sample points in the event.

Probability of event A

Assigning Probabili)es: How?

1. Classical Method o Assigning probabili)es based on the assump)on of equally likely outcomes

2. Rela)ve Frequency Method o Assigning probabili)es based on historical data

3. Subjec)ve Method o Assigning probabili)es based on judgment

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Probability rules for sample points 1) individual probabili)es must lie between 0 and 1 (inclusive) 2) the sum of probabili)es of all sample points in a sample space must equal 1

Classical Method

If an experiment has n possible outcomes, the classical method would assign a probability of 1/n to each outcome.

o Experiment: rolling a die o Sample Space: S = {1, 2, 3, 4, 5, 6} o Probabili)es:

Each sample point has a 1/6 chance of occurring

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Rela)ve Frequency Method

Rela)ve frequency of an event =

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# of sample points in the event

Total # of sample points in the historical data

Lucas Tool Rental would like to assign probabili)es to the number of car polishers it rents each day. Oce records show the following frequencies of daily rentals for the last 40 days.

Number of polishers Rented

Number of days Probability

0 5 5/40 = 0.125

1 15 15/40 = 0.375

2 20 20/40 = 0.5

Total 40 1

Subjec)ve Method Express our degree of belief that the experimental outcome will occur Tim and Judy just made an oer to purchase a house. Two outcomes are possible: E1 = their oer is accepted E2 = their oer is rejected

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Judy believes the probability their oer will be accepted is 0.8 Judy would set P(E1) = 0.8 and P(E2) = 0.2. Tim, however, believes the probability that their oer will be accepted is 0.6 Tim would set P(E1) = 0.6 and P(E2) = 0.4.

What if?

Can you u)lize the deni)ons/concepts for probability, experiment, outcome, and event?

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Example Experiment: Toss a fair coin twice and note faces. What is the sample space for this experiment? What is the probability of ge`ng two tails?

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Example

In an opaque bag, there are 5 green M&Ms, 2 red M&Ms, and 3 blue M&Ms.

If you randomly pick 1 M&M from the bag, what is the probability that

a) you get a green M&M? b) you get a blue M&M? c) you get a red M&M?

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Example The table below provides probabili)es for the number of

calls received by the Marke)ng main oce during lunch )me (12:00 1:00 PM) on a given workday. The oce receives at most 4 calls during lunch )me per day.

1. P(3 calls) = ? 2. P(At least 1 call during lunch )me) = ?

# of calls 0 1 2 3 4 Probability 0.1 0.4 0.2 0.1

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Take Aways

1) Probability is always between 0 and 1, inclusive. 2) The Sum of the probabili)es of all sample points in an

experiment is 1. 3) Can you explain how to assign probabili)es? 4) What do the following terms mean? Experiment, outcome (sample point), sample space, event

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