Chapter

Five

McGraw-Hill/Irwin

© 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

A Survey of Probability A Survey of Probability ConceptsConcepts

Descriptive Statistics deals with what happened in the Descriptive Statistics deals with what happened in the pastpast..

Inferential Statistics deals with what may happen in the Inferential Statistics deals with what may happen in the futurefuture..

In studying about future events, we begin with the concept of In studying about future events, we begin with the concept of probabilityprobability..

Expressed as a decimal between 0 and 1Expressed as a decimal between 0 and 1

Classical ProbabilityClassical Probability(assumes all outcomes are equally likely)

Probability of an event = Probability of an event =

No of possible favorable outcomes No of possible favorable outcomes Total number of possible outcomesTotal number of possible outcomes

Eg.Eg. When you throw a die, what is P(Even) ?When you throw a die, what is P(Even) ? Possible even numbers you can get are: 2, 4 & 6 (3 possible favorable outcomes)Possible even numbers you can get are: 2, 4 & 6 (3 possible favorable outcomes)Total possible outcomes are: 6 (ie. You can get 1,2,3…6)Total possible outcomes are: 6 (ie. You can get 1,2,3…6)

So, P(Even) = 3/6 = 0.5So, P(Even) = 3/6 = 0.5

You can think of card games, lotteries as examples of classical probability.You can think of card games, lotteries as examples of classical probability.

Events are Mutually ExclusiveMutually Exclusive if the occurrence of any one event means that none of the others can occur at the same time.

Mutually exclusive: Rolling a 2 precludes rolling a 1, 3, 4, 5, 6 on the same roll.

Events are Collectively ExhaustiveCollectively Exhaustive if at least one of the outcomes must occur when an experiment is conducted.

If you throw a die, you must get either 1 or 2 or…6; ie. one of the events must occur.

Rules for Computing Rules for Computing ProbabilitiesProbabilities (Memorize, ie. thru (Memorize, ie. thru understanding!)understanding!)

Complement rule: P(A) = 1 – P(~A)Complement rule: P(A) = 1 – P(~A)

Special rule of addition: P(A or B) = P(A) + P(B)Special rule of addition: P(A or B) = P(A) + P(B)General rule of addition: P(A or B) = P(A) + P(B) – P(A & B)General rule of addition: P(A or B) = P(A) + P(B) – P(A & B)

Special rule of multiplication: P(A and B) = P(A)xP(B)Special rule of multiplication: P(A and B) = P(A)xP(B)General rule of multiplication: P(A and B) = P(A)xP(B|A)General rule of multiplication: P(A and B) = P(A)xP(B|A) or = P(B)xP(A|B)or = P(B)xP(A|B)

No. of combination from n objects taken r at a time:No. of combination from n objects taken r at a time:nCr = n! / [r! (n-r)!]nCr = n! / [r! (n-r)!]

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

If two events A and B are mutually

exclusive, theProbability of A or B

equals the sum of their respective probabilities.

Special Rule of Addition

Checkout problem Checkout problem #14 (a) on page 133#14 (a) on page 133

A~~AA

The Complement rule using a Venn Diagram

Set of all possible outcomesSet of all possible outcomes

A subset of outcomesA subset of outcomesThe possible The possible remaining outcomesremaining outcomes

~A is ~A is called the called the complemecomplement of Ant of A

P(A) + P(~A) = 1; thus, P(A) = 1 – P(~A)

Checkout problem Checkout problem #14 (b) on page 133#14 (b) on page 133

If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:

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

The General Rule of Addition

Joint probability

What is the probability a tourist visited Disney World OR Busch Gardens?What is the probability a tourist visited Disney World OR Busch Gardens?

P(Disney OR Busch) = P(Disney) + P(Busch) – P(Disney AND Busch)P(Disney OR Busch) = P(Disney) + P(Busch) – P(Disney AND Busch)

= .6 + .5 - .3 = .8= .6 + .5 - .3 = .8

Venn diagram

The area that is The area that is ‘double-counted’ ‘double-counted’

must be subtracted.must be subtracted.This is what the This is what the formula does.formula does.

Note:If a tourist is allowed to visit only one park (ie mutually exclusive), then P(Disney and Busch)=0. Then, the general rule becomes the special rule!

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

Events A and B must be independent.Independence means occurrence of one event has no effect on the probability of the occurrence of the other.

The Special Rule of MultiplicationSpecial Rule of Multiplication

Eg.You toss 2 coins (or a coin two times). What is P(2 Heads)?P(2Heads) = P(I coin is head) x P(II coin is also head)Note that the outcome of one coin does not influence the other.

So, P(2 Heads) = .5 x .5 = .25

Events A and B are not independent.

ie. One event has effect on the probability of the occurrence of the other.

The rule is written: P(A and B) = P(A)P(B|A) or P(A and B) = P(B)P(A|B)

The General Rule of MultiplicationGeneral Rule of Multiplication

Notes:The probability of event A occurring given that the event B has occurred is written P(A|B). It is called a Conditional probability.If A and B are independent, P(A|B) = P(A). Think about a coin toss expt. Whether you get a H or T is not dependent on the outcome of the prior toss.

Question 1: If a student is selected at random, what is the probability the student is a female?

P(F) = 400/100 (see Table in prior slide)

Question 2: What is the probability, the student is a female accounting major?

P(F and A) = 110/1000 (again, see prior Table)

Question 3: Given that the student is a female, what is the probability that she is an accounting major? ie Find P(A|F)

We know from the General rule that P(F and A) = P(F) x P(A|F)

Re-arranging the formula,P(A|F) = P(F and A) / P(F)

= [110/1000] / [400/1000]= .275

Major Male Female Total

Accounting 170 110 280

Finance 120 100 220

Marketing 160 70 230

Management 150 120 270

Total 600 400 1000

Convenient

Visits Yes No Total

Often 60 20 80

Occasional

25 35 60

Never 5 50 55

90 105 195

1.001.00

A Tree DiagramTree Diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages.

Based on Self-Review 5-8 Page 140Based on Self-Review 5-8 Page 140

)!(!

!

rnr

nCrn

A CombinationCombination is the number of ways to choose r objects from a group of n objects where order does not matter.

where, n! = 1x2x3x ..... (n-2)x(n-1)xnwhere, n! = 1x2x3x ..... (n-2)x(n-1)xn

You want to select a committee of two out of three people. How many You want to select a committee of two out of three people. How many ways are possible?ways are possible?

33CC22 = 3! / [2! (3-2)!] = 3 ways = 3! / [2! (3-2)!] = 3 ways

Say, these three people are: Mr.A, Mrs.B and Ms.C.Say, these three people are: Mr.A, Mrs.B and Ms.C.The three possible committee combinations are: A,B or A,C or B,CThe three possible committee combinations are: A,B or A,C or B,C