# MIS 446 Lecture Notes - Lecture 7: Binomial Distribution, Continuous Or Discrete Variable, Fair Coin

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14 Sep 2016

School

Department

Course

Professor

MIS 446 Advanced Business Analytics: Probability and Sampling Distributions

Agenda

oProbability and Probability Distributions

oSampling Distributions

oConfidence Interval Estimation

oSample Size Determination

Basic Probability Concepts

oProbability – the chance that an uncertain event will occur (always between 0 and 1)

oImpossible Event – an event that has no chance of occurring (probability = 0)

oCertain Event – an event that is sure to occur (probability = 1)

Events

oEach possible outcome of a variable is an event.

Simple event

An event described by a single characteristic

e.g., A day in January from all days in 2013

Joint event

An event described by two or more characteristics

e.g. A day in January that is also a Wednesday from all days in 2013

Complement of an event A (denoted A’)

All events that are not part of event A

e.g., All days from 2013 that are not in January

Probability Examples

oSuppose we roll 2 dice

Probability die rolls sum to three = 2/36

oSuppose two consumers try a new product.

Assume equally likely possible outcomes:

1. like, like

2. like, dislike

3. dislike, like

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4. dislike, dislike

oProbability at least one dislikes product = 3/4

Probability Summary

oProbability is the numerical measure of the likelihood that an event will occur

oThe probability of any event must be between 0 and 1, inclusively

oThe sum of the probabilities of all mutually exclusive and collectively exhaustive events

is 1

General Addition Rule

o

oIf A and B are mutually exclusive, then P(A and B) = 0, so the rule can be simplified:

o

o

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Review: Discrete vs Continuous Variables

oDiscrete variables produce outcomes that come from a counting process (e.g. number of

classes you are taking).

oContinuous variables produce outcomes that come from a measurement (e.g. your

annual salary, or your weight).

Discrete Random Variables

oCan only assume a countable number of values

oExamples:

Roll a die twice

Let x be the number of times 4 occurs (then x could be 0, 1, or 2 times)

Toss a coin 5 times

Let X be the number of heads (then X = 0, 1, 2, 3, 4, or 5)

oAn example of Discrete random variable probability distribution

oDiscrete Variables Expected Value

Expected Value (or mean) of a discrete variable (Weighted Average)

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