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Unformatted text preview: Business Statistics
Fifth Edition Ken Black Chapter 4: Probability Not to be reproduced without the written permission of F .8, Alt 4—1 4.1 Introduction to Probability (page 98)
o Synonyms for probability: , 0 Probability is the language of statistics.
0 “An experiment is a process that produces outcomes.” (page 100)
0 We will only consider random experiments (page 100) 0 Characteristics of random experiments: I Can specify all possible outcomes; and, I An outcome cannot be predicted with certainty. Examgle: Today’s closing price of a security relative to yesterday. P .
OSSIbIC Which one will occur? outcomes 0 Probability measures the uncertainty of the outcomes. 0 Chapters 4—6 look at ways of dealing with and modelling uncertainty. Not to be reproduced without the written permission of PB. Alt 42 4.2 Methods of Assigning Probabilities o The classical method (page 99)
o N is the total number of possible mutually exclusive equally likely outcomes.
0 The probability of an event E equals the ratio of the number of outcomes (n5) pertaining to E to the total number of outcomes (N): P(E) = nE [N
Examgle:
Roll a fair die one time and observe the up face.
P (rolling a 6) = Roll a pair of fair dice and observe the up faces P(both up faces are 6’s) = = { Why? Examgle: Each season, Anand’s new fashion designs are shown to a panel
of experts  ten R.H.Smith undergraduates — and each secretly votes thumbs up or down. Here are historical data on the panel’s rankings of 1500 designs
and the subsequent market success, measured by sales: Number of Positive Panel Votes Sales 710 4—6 03 Total Very successful(V) 380 140 80 600 Modest successful(M) 180 120 100
Disa ointment 40 40 420 Total 600 300 600 1500 What is the probability that a randomly chosen design will be given 7~10 thumbs up? Not to be reproduced without the wn'tten permission of F .8. Alt 4~3 o The Relative Frequency of Occurrence (page 99) o “. . .the probability of an event is equal to the number of times the event has
occurred in the past divided by the total number of opportunities for the event to have occurred.” (page 99) 0 Assume an experiment can be repeated indeﬁnitely under identical conditions. The probability of an event E is the long—run relative frequency with which
the event E occurs. P(E) = limit (relative frequency) as number of trials 9 00 Example: Flipping a coin to ﬁnd the P(Head) Number of heads Relative Frequency
3 out of 5 0.5 in each of 5 tosses 3 out of 5
2 out of 5,
Etc. Not to be reproduced without the written permission of F.B. Alt 44 0 Subjective Probability 0 “Based on the feelings or insights of the person determining the probability.”
0 A way of assigning subjective probabilities Examgle: What is the probability that the Colts will play in the Super
Bowl in 2013? Suppose you say Event (Colts Will Play in Super Bowl) Event (Pick red bead)
P(Colts Play in Super Bowl) = P(Pick red bead) = DEEDS
DEEDS Are you indiﬂerent between wagering on either event? If so, then P(Colts will play in Super Bowl in 2013) =
If not, then P(Colts will play in Super Bowl in 2013) i
o Regardless of how you assign probabilities, OsP(E)sl for any event Not to be reproduced without the written permission of FB. Alt 45 o 4.5 Addition Laws 0 Special Law of Addition (page 112) I Let X and Y denote any events in a random experiment. I If X and Y are mutually exclusive events (cannot occur simultaneously), P(X or Y) = P(X) + P(Y) S
Venn
diagram
Example:
Number of Positive Panel Votes
Sales 710 4—6 03 Total Very successful(V) 380 140 80
Modest successful(M) 180 120
Disa ointment D 40 40 Total 600 300 600 1500 600 What is the probability that a randomly chosen design will be given “7—10”
or “46” thumbs up? 13(“7—10” or “46”) = Not to be reproduced without the written permission of RB Alt 4—6 0 General Law of Addition (page 107) P(X or Y) = P(X) + P(Y) — P(X and Y) X and Y has been counted twice so subtract once. Examgle:
Number of Positive Panel Votes
Sales 710 46 03 Total Very successful(V) 380 140 80
Modest successful(M) 180 120
Disa ointment D 40 40 Total 600 300 600 1500 600 What is the probability that a randomly chosen design will be given 710
thumbs up or be “Very Successful”? P(“Very Successful” or “710” ) = Not to be reproduced without the written permission of RB. Alt 47 0 Complementary Events (page 103)
0 Not A is called the complement of A. 0 Notation : A'
0 Law: P(not A) = 1 — P(A),
or P(A) = 1 — P(not A) Example (Birthday Problem): There are 50 people in a room. What is the probability that at least two of them have the same birthday? Assumptions:
N0 birthdays fall on Feb 29; All 365 days are equally likely for each person; and
Birthdays are independent A = {at least two people have same birthday} A, = { people have the same birthday}
P(A ')= = forn=50
H P (A) =
_I_l_ P(A)
23 51
3O 71
50 Not to be reproduced without the written permission ofFB. Alt 4~8 4.7 Conditional Probability (page 120) o The conditional probability of event X occurring, given that event Y has occurred, is denoted by P(XY) where:
P(XY) = P(X and Y) / P(Y), provided P(Y) > 0. X and Y
Example:
Number of Positive Panel Votes
Sales 710 46 0—3 Total Very successful(V) 380 140 80
Modest successful(M) 180 120 100 Disa ointmentD 40 40 420
Total 600 300 600 1500 What is the probability that a randomly chosen design will be
“Very Successful” given that it was given “7—10” thumbs up? P(Very Successful I 7—10): P(VSl7—10)= W = Note: P (v3) = .40 while P(VSI 710) = Not to be reproduced without the written permission of F.B. Alt 49 4.6 Multiplication Laws 0 From the deﬁnition of conditional probability, P(X and Y) = P(Y) P(XIY)
Obviously,
P(X and Y) = P(X) P(Y IX) Examgle: Randomly pick (without replacement) 2 cards from a standard
deck. Find probability of 2 hearts. X = {ISt card is a heart}, Y = {2nd card is a heart} P (X and Y) = P(X) P(YIX) = o Multiplication law useful in Probability Trees (page 129) Not to be reproduced without the written permission of F .3 Alt 410 0 Probability Trees (page 129)
o In a probability tree, the probability for a Speciﬁc path is found by
using the multiplication rule.
Example: A purchasing dept ﬁnds that 75% of its special orders are received
on time. Of those orders that are on time, 80% meet speciﬁcation completely;
of those orders that are late, 60% meet them. T = {Order is on time} M = {Meets speciﬁcations}
P (T) = .75 P(MlT) = .80 P (M! T’) = .60 PM”) 2 '80 P(T and M) = .60 P(T) = .75 P(T andM’)=.15 P(Ml T’) =.60 P(T'and M): .15 P(T'andM')= .10 a. Find the probability that an order is on time and meets speciﬁcations. P(T and M) = b. Find the probability that an order meets speciﬁcations. P(M) = PL_~_) + Pt_____) II
+
H Not to be reproduced without the written permission of F.B. Alt 4! l 0 Independent Events (page 123) o X and Y are independent events if and only if P(XY) = P(X) Example: Are events “Very Successful” and “7 — 10” independent?
P(VSl7—10)= 380/600 = .63 (from page 49)
P(VS) = 600/1500 = .40 Conclusion: The events “VS” and “710” (are, are not) independent because P(VSl710) ( = , 75) P(VS). o Multiplication Law for Independent Events (page 117) If X and Y are independent, then P(X and D = P(X) P(Y) Reason: From Multiplication Law,
P(X and Y) = P(XIY) P(Y).
From Independence,
P(XlY) = P(X).
By substitution,
P(X and Y) = P(X) P(Y). Example: Roll 3 pair of fair dice and observe the up faces. P(both up faces are 6’s) = { Why? Refer to page 4—3. Not to be reproduced without the written permission of PB. Alt 412 0 Summary of Chapter 3
o 3 Interpretations of Probability
I Classical approach (equally likely outcomes)
I Relative frequency approach
I Subjective approach
0 P(not A) = l — P(A)
0 Addition Law
Are events A and B mutually exclusive?
If yes: P(X or Y) = P(X) + P(Y)
If no: P(X or Y) = P(X) + P(Y) — P(X and Y)
o Conditional probability
P(X  Y) = P(X and Y) / P(Y)
o Multiplication Law
P(X and Y) = P(X  Y) P(Y) {Useful in probability trees.
0 Determining Independence of events X and Y
I If P(X  Y) = P(X), then X and Y are independent events.
I If X and Y are independent events, then P(X and Y) = P(X) P(Y) Not to be reproduced without the written permission of PB. Alt 4‘13 ...
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 Conditional Probability, Probability, Probability theory, Bayesian probability, written permission

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