Chapter 3

Probability

BUSINESS STATISTICS

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Learning Objectives

In this chapter, you learn:

● Basic probability concepts and definitions● Conditional probability ● To use Bayes’ Theorem to revise probabilities● Some Problems

INTRODUCTION

In our day to day life, we may face many situations where uncertainty plays a vital role. We usually use statements like ”there is a chance for rain today” or ”probably i will get A grade in university examination” etc. In all these contexts the term chance or probably is used to indicate uncertainty.

The word Probability is related with the occurrence of uncertainty, and Probability theory is the discipline which tries to quantify the concept of chance or likelihood.

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Important Terms

Random Experiment : An Experiment is said to be Random Experiment if the outcome cannot be predicted in advance eventhough we know all the possible outcomes .

Examples: 1.Tossing a coin once is a random experiment ,since we know that there are two outcomes head and tail, but we cannot say which will come up while tossing. 2.Throwing a die is a R.E,since any one of the six numbers 1,2,3,4,5,6 may appear

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Sample Space

The Sample Space is the collection of all possible events

e.g. All 6 faces of a die:

S= {1,2,3,4,5,6}

e.g. All 52 cards of a bridge deck:

S= {1,2,3...56}

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Events

Event: Each Possible outcome of a variableSimple events:

■ An outcome from a sample space with one characteristic■ e.g., A red card from a deck of cards

Complement of an event A (denoted A’)■ All outcomes that are not part of event A■ e.g., All cards that are not diamonds

Joint event■ Involves two or more characteristics simultaneously■ e.g., An ace that is also red from a deck of cards

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Visualizing Events

■ Venn Diagrams■ Let A = aces■ Let B = red cards

A

B

A ∩ B = ace and red

A U B = ace or red

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Mutually Exclusive Events

■ Mutually exclusive events■ Events that cannot occur together ie., the

occurrence of one excludes the other

Example: A= queen of diamonds; B = queen of clubs

■ Events A and B are mutually exclusive

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Collectively Exhaustive Events

● Collectively exhaustive events○ One of the events must occur ○ The set of events covers the entire sample space

Example: A = aces; B = black cards;

C = diamonds; D = hearts

○ Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace may also be a heart)

○ Events B, C and D are collectively exhaustive and also mutually exclusive

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Mutually Independent events

Two events are said to be independent if the occurrence of one event has no influence over the occurrence of the other.

Example: Consider the experiment of tossing two coins.In this the event getting head in the first coin will not influence the event getting head in the second coin, and therefore it is independent.

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Assessing Probability

■ There are three approaches to assessing the probability of an uncertain event:1. a priori classical probability

2. Empirical classical probability

3. subjective probability an individual judgment or opinion about the probability of occurrence

CLASSICAL DEFINITION

Eventhough there are different approaches in defining probability. The oldest and simplest one is classical definition or mathematical definition.

Definition: If a random experiment results in n exhaustive, mutually exclusive and equally likely cases, m of them are favourable (m ≤ n) to the occurrence of an event A, then the probability of A, denoted as P(A), is defined as

P(A) = m / n

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Limitations Of Classical Definition

Classical definition of probability is very easy to understand. But the definition may not be applicable in all situations. Following are some of the limitations of classical definition of probability. ● If the events cannot be considered as equally likely,

classical definition fails. ● When the total number of possible outcomes n become

infinite this definition cannot be applied.

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Axiomatic approach of Probability

Axiomatic definition of probability was introduced by Russian mathematician A.N. Kolmogrov and it approaches probability as a measure. Axioms of Probability: Let A be an event . Then theProbability of the event A denoted by P(A) is a real number satisfying the following conditions.● P(A) ≥ 0 .● P(S)=1 .[Total Probability =1]Ą● If A1, A2,... is a sequence of mutually exclusive events in S then,

P(A1 U A2 U A3 U….UAn) = P(A1) + P(A2) +......P(An)

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Certain

Impossible

0.5

1

0

Theorems on Probability

● Probability of an impossible event is 0. Ie P(Φ) = 0● Probability of the complementary event A namely Ā is given

by , P(Ā) = 1- P(A).● For any two events A and B , P(Ā ∩ B) = P(B) - P(A ∩ B)● If A and B are two events such that A ⊂ B then, P(B ∩ A’) = P(B) - P(A).

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General Addition Rule

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

General Addition Rule: If A and B are any two events and are not disjoint , then

COROLLORY

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

● For three non -Mutually Exclusive events A,B and C We have,

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P(AUBUC) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) -P(C∩A) + P (A∩B∩C)

P(AUB) = P(A) + P(B)

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General Addition Rule Example

P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace)

= 26/52 + 4/52 - 2/52 = 28/52Don’t count the two red aces twice!

BlackColor

Type Red Total

Ace 2 2 4Non-Ace 24 24 48Total 26 26 52

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Conditional Probabilities

■ A conditional probability is the probability of one event, given that another event has occurred:

Where P(A and B) = joint probability of A and B P(A) = marginal probability of A ≠ 0

P(B) = marginal probability of B ≠ 0

The conditional probability of A given that B has occurred

The conditional probability of B given that A has occurred

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What is the probability that a car has a CD player, given that it has AC ?

i.e., we want to find P(CD | AC)

Conditional Probability Example

■ Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

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Conditional Probability Example

No CDCD TotalAC 0.2 0.5 0.7No AC 0.2 0.1 0.3Total 0.4 0.6 1.0

■ Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

(continued)

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Statistical Independence

■ Two events are A and B are said to be independent events if and only if:

Events A and B are independent when the probability of one event is not affected by the other event. This is also called as Multiplication law of probability when the events are independent.

P(A∩B) = P(A).P(B)

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Multiplication Rules

■ Multiplication rule for two events A and B: P(A∩B) = P(A).P(B∣A), P(A) > 0P(A∩B) = P(B).P(A∣B), P(B) > 0

Where P(B∣A) represent conditional probability of occurrence of B when the event A has already happened and P(A∣B) represents conditional probability of occurrence of A when the event B has already happened.Note:This is called as Multiplication theorem of probability when the events are dependent.

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Bayes’ Theorem

Statement: If E1,E2,E3...En are Mutually disjoint events with P(Ei)≠ 0 (i=1,2,...n) then for any arbitrary event A associated with Ei.then,

P(Ei|A) =P(Ei).P(A|Ei) / Σ P(Ei).P(A|Ei)

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Bayes’ Theorem Example

■ A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test,

what is the probability that the well willbe successful?

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■ Let S = successful well U = unsuccessful well

■ P(S) = 0.4 , P(U) = 0.6 (prior probabilities)■ Define the detailed test event as D■ Conditional probabilities:

P(D|S) = 0.6 P(D|U) = 0.2■ Goal is to find P(S|D)

Bayes’ Theorem Example(continued)

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Bayes’ Theorem Example(continued)

Apply Bayes’ Theorem:

So the revised probability of success, given that this well has been scheduled for a detailed test, is 0.667

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■ Given the detailed test, the revised probability of a successful well has risen to 0.667 from the original estimate of 0.4

Bayes’ Theorem Example

Event PriorProb.

Conditional Prob.

JointProb.

RevisedProb.

S (successful) 0.4 0.6 (0.4)(0.6) = 0.24 0.24/0.36 = 0.667

U (unsuccessful) 0.6 0.2 (0.6)(0.2) = 0.12 0.12/0.36 = 0.333

Sum = 0.36

(continued)

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Chapter Summary

■ Discussed basic probability concepts■ Sample spaces and events, contingency tables, simple

probability, and joint probability■ Examined basic probability rules

■ General addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive events

■ Defined conditional probability■ Statistical independence, marginal probability and the

multiplication rule■ Discussed Bayes’ theorem.