ch. 4b narrated power points

CHAPTER FOUR-B: 

CHAPTER FOUR-B INTRODUCTION TO PROBABILITY (cont.)

4.4 Conditional Probability: 

4.4 Conditional Probability Previous examples were simultaneous examples. Both companies in our investment example, Markley and Collins, operated at the same time and probability was examined simultaneously. In conditional probability, one event occurs before a second event, and we have reason to believe or know that being aware of the occurrence of the first event changes the probability of the second event. Or, knowing one fact changes the probability of the second fact being true or false.

Conditional Probability: 

Conditional Probability What are some examples where we would expect conditional probability? Drawing a card from a deck and not replacing the card drawn; if we draw a king with the first draw, does the probability of drawing a second king change? Financial markets: we often see that when the stock market does well, the bond market falls. We need a roll of 10 in Monopoly to get past Go; the first die lands 3 and the second die is still spinning; what can we say about our chances of collecting $200? What about our chances of landing short of Go?

Slide 4: 

The new probability of an event given that another related event has already occurred is called a conditional probability . A conditional probability is computed as follows : The conditional probability of A given B is denoted by P ( A | B ). 4.4 Conditional Probability No longer talking simultaneous; now SEQUENCE.

Conditional Probability (cont.): 

Conditional Probability (cont.) Formula: P ( Conditional event |First event) = P ( Conditional event  First event) P ( First event) That is … P (B | A) = P (B  A) P (A) REM: this formula assumes that knowing A affects the probability of B, it doesn’t prove that A affects B. We are saying that the probability of B happening changes when A occurs (and we assume A’s occurrence has an affect on event B, or knowing A’s occurrence helps us with the probability of B) is calculated using this formula.

Slide 6: 

Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M  C ) = .36, P ( M ) = .70 Thus: Conditional Probability = Collins Mining Profitable given Markley Oil Profitable Note: P ( M  C ) = P ( C  M )

Conditional Probability: Bettering the Example: 

Conditional Probability: Bettering the Example Your textbook uses an example about promotions within a police department. Here, I am changing the facts for better understanding (I hope). The joint probability table is the same, and a historical record of the past practices for the department. At a special meeting, the chief says “I’m pleased to announce a promotion.” Steve and Sue both applied, along with many other officers on the force, and look at each other thinking “could it be me?” They have a .24 and .03 probability of being selected respectively. See the probability table:

Conditional Probability: Police Promotion Example: 

Conditional Probability: Police Promotion Example Men Women Total Promoted 288 36 324 Not promoted 672 204 876 Total 960 240 1200 Now figure the independent probabilities based on past experience of all 1200 promotions: Men(M) Women (W) Total Promoted (A) .24 = 288/1200 .03 .27 Not promoted (A c ) .56 .17 .73 Total .80 .20 1.00 This is a joint probability table; the probabilities of two events. The probabilities in the margins are the marginal probabilities found by adding the probabilities in the row and column.

Slide 9: 

P( A c | M ) = ? P( M ) = .8 P( W ) = .2 P( A | W ) = ? P( A c | W ) = ? P( A | M ) = ? P( M  A ) = .24 P( W  A ) = .03 P( W  A c ) = .17 P( M  A c ) = .56 Police Promotions Tree Diagram: Conditional Probabilities Gender Promotion given gender Experimental Outcomes

Conditional Probability: Police Promotion Example: 

Conditional Probability: Police Promotion Example P (M) = .80 P (W) = .20 P (A) = .27 P ( A C ) = .73 P (M  A) =.24; P (M  A c ) =.56; P (W  A) =.03; P (W  A C ) =.17 The chief then says, “This new captain has been an exemplary officer since he joined the force.” Have Steve’s chances improved that he will be named? P (A | M) = P (A  M)/ P (M) = .24/.80 = .30 (up from .24) Chief: “Oops … I mean since she joined the force.” Have Sue’s chances improved that she will be named? P (A | W) = P (A  W)/ P (W) = .03/.20 = .15 (up from .03)

Slide 11: 

P( A c | M ) = .70 P( M ) = .8 P( W ) = .2 P( A | W ) = .15 P( A c | W ) = .85 P( A | M ) = .30 P( M  A ) = .24 P( W  A ) = .03 P( W  A c ) = .17 P( M  A c ) = .56 Police Promotions Tree Diagram: Conditional Probabilities Gender Promotion given gender Experimental Outcomes

Slide 12: 

Independent Events If the probability of event A is not changed by the existence of event B , we would say that events A and B are independent . Two events A and B are independent if: P ( A | B ) = P ( A ) P ( B | A ) = P ( B ) or Events are considered to be independent if, using our formula for conditional probability, the probability of event B causing event A = the probability of event A. That is, the probability of event B happening has no effect on the probability of event A happening.

Slide 13: 

P( T | H ) = .5 P( H ) = .5 P( T ) = .5 P( H | T ) = .5 P( T | T ) = .5 P( H | H ) = .5 P( H  H ) = .25 P( T  H ) = .25 P( T  T ) = .25 P( H  T ) = .25 Independent Events: Coin Flip Tree Diagram: Conditional Probabilities First Coin Second Coin Experimental Outcomes

Independent Events: 

Independent Events Given that I have drawn a red card, what is the probability that I have drawn a king? P (King | Red) = P (Red  King) = (2/52) P (Red) (26/52) = .077 (1/13 th ) What’s the probability of drawing a king to begin with? P (King) = 4/52 = .077 (1/13 th ) These are independent events because the probability did not change. Compare: I’ve drawn a king; what is the probability that I will draw a second king next? Ans : 3/51. (These are not independent events.)

Slide 15: 

Multiplication Law: Used to calculate the intersection The multiplication law provides a way to compute the probability of the intersection of two events. The law is written as: P ( A  B ) = P ( B ) P ( A | B ) Whats going on here? We are going to use the formula for conditional probability to determine the intersection of two events, A and B, by multiplying each side by P(B). If the intersection number is not given to us, we can calculate it IF we know the P(B) and the P(A|B).

Slide 16: 

Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M ) = .70, P ( C | M ) = .5143 Multiplication Law M  C = Markley Oil Profitable and Collins Mining Profitable Thus: P ( M  C) = P ( M ) P ( M|C ) = (.70)(.5143) = .36 (This result is the same as that obtained earlier using the definition of the probability of an event.)

Slide 17: 

The multiplication law also can be used as a test to see if two events are independent. The law is written as: P ( A  B ) = P ( A ) P ( B ) Multiplication Law for Independent Events If we know that two events are independent, we can easily calculate the intersection of these two events by multiplying their individual probabilities. Also, we can use this formula to determine if events are independent; if the probability of their intersection is different from the product of their individual probabilities, then we know the are not independent events.

Slide 18: 

Multiplication Law for Independent Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable We know: P ( M  C ) = .36, P ( M ) = .70, P ( C ) = .48 But: P ( M)P(C) = (.70)(.48) = .34, not .36 Are events M and C independent? Does  P ( M  C ) = P ( M)P(C) ? Hence: M and C are not independent. Independent: if A happens, it has no affect on B; both happen. Mutually exclusive: when A occurs, B doesn’t occur; or events A and B can never happen together.

Guide to Selecting Probability Rules: 

Guide to Selecting Probability Rules Joint Events Complementary ( not A) Union (A or B) Intersection (A and B) Conditional (A given B) Rule of Complements Addition Rule Multiplication Rule Conditional Rule P(A C )= 1 – P(A) Mutually Exclusive P(A u B)= P(A)+P(B) Not mutually exclusive P(A u B)= P(A)+P(B)- P(A ∩ B) Independent P(A ∩ B) = P(A) ∙ P(B) Dependent P (A ∩ B) = P(A|B)∙P(B) Independent P(A|B)=P(A) P(B|A)=P(B) Dependent P(A|B)= P(A∩B) P(B) P(B|A)= P(A∩B) P(A)

4.5 Bayes’ Theorem: 

4.5 Bayes’ Theorem New Information Application of Bayes’ Theorem Posterior Probabilities Prior Probabilities Often we begin probability analysis with initial or prior probabilities . Then, from a sample, special report, or a product test we obtain some additional information. Given this information, we calculate revised or posterior probabilities . Bayes’ theorem provides the means for revising the prior probabilities.

Bayes’ Theorem: 

Bayes’ Theorem What are we doing here? We are going to be playing with the formula for conditional probability, P (B | A) = P (B  A) P (A) to solve for P ( A | B) which = P (A  B) P (B) If P (A | B) = P (A  B)/ P (B) Then P (A  B) = P(B) P (A | B) (that’s just algebra) And P (A  B) = P (B  A) And P (B) = P (A 1  B) + P (A 2  B) when P (A 1 ) + P (A 2 ) = 1 (they are complements) (We are going to replace unknowns with knowns .)

Slide 22: 

A proposed cancer medication may help patients suffering from a particular form of cancer. The drug cannot move to human trials until after trials on lab rats is completed. If there are positive results with the lab rats, then the drug can be tried on human patients. Past history of drug testing shows that a new drug works on patients 70% of the time, and does not work on human patients 30% of the time. Bayes’ Theorem Example: PharMor Pharmecuticals

Slide 23: 

Prior Probabilities Let: Bayes’ Theorem A 1 = drug does work on human patients A 2 = drug does not work on human patients P( A 1 ) = .7, P( A 2 ) = .3 Using subjective judgment: REM: testing on lab rats first, then move to human trials.

Bayes’ Theorem: 

New Information The testing on the lab rats has succeeded; lab rats with this form of cancer show positive results from taking the new drug. Let B denote the event of a positive result of the drug on lab rats. Given that B has occurred, should PharMor revise the probabilities that the new drug will work on humans? That is, what is P (A 1 | B) ? Note: the testing on lab rats and the testing on humans are mutually exclusive events; they happen separately. We see, in the next slide, that they are not independent, because when the lab rats respond one way, human beings respond differently. Bayes’ Theorem

Slide 25: 

Conditional Probabilities Past history with new medications indicate the following: Bayes’ Theorem P ( B | A 1 ) = .8 P ( B | A 2 ) = .1 P ( B C | A 1 ) = .2 P ( B C | A 2 ) = .9 Hence: B = drug works on lab rats; and B C = the complement of drug works on lab rats (i.e. drug does not work on rats) A 1 = does work on humans; A 2 = does not work on humans We want to know P (A 1 | B)

Slide 26: 

P( B c | A 1 ) = .2 P( A 1 ) = .7 P( A 2 ) = .3 P( B | A 2 ) = .1 P( B c | A 2 ) = .9 P( B | A 1 ) = .8 P( A 1  B ) = .56 P( A 2  B ) = .03 P( A 2  B c ) = .27 P( A 1  B c ) = .14 Bayes’ Theorem Tree Diagram Humans Lab rats Experimental Outcomes

Bayes’ Theorem: 

Bayes’ Theorem To find the posterior probability that event A i will occur given that event B has occurred, we apply Bayes’ theorem . Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space. Another way of thinking of posterior probabilities is updating prior probability values after an experiment.

Bayes’ Theorem: 

Posterior Probabilities Given that the drug has worked on lab rats, we revise the prior probabilities as follows: Bayes’ Theorem = .95 __ (.7)(.8)___ (.7)(.8) + (.3)(.1)

Slide 29: 

Conclusion The fact that the new medication worked on the lab rats is good news for PharMor . The posterior probability of the medication working on human beings is .95 compared to a prior probability of .70. Bayes’ Theorem

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach Step 1 Prepare the following three columns: Column 1 - The mutually exclusive events for which posterior probabilities are desired. Column 2 - The prior probabilities for the events. Column 3 - The conditional probabilities of the new information given each event.

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach (1) (2) (3) (4) (5) Events A i Prior Probabilities P ( A i ) Conditional Probabilities P ( B | A i ) A 1 A 2 .7 .3 1.0 .8 .1

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach Step 2 Column 4 Compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, P ( A i I B ) = P ( A i ) P ( B | A i ).

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach (1) (2) (3) (4) (5) Events A i Prior Probabilities P ( A i ) Conditional Probabilities P ( B | A i ) A 1 A 2 .7 .3 1.0 .8 .1 .56 .03 Joint Probabilities P ( A i I B ) .7 x .8

Slide 34: 

Bayes’ Theorem: Tabular Approach Step 2 (continued) We see that there is a .56 probability of the drug working on humans and working on lab rats. There is a .03 probability of the drug not working on humans but working on lab rats.

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach Step 3 Column 4 Sum the joint probabilities. The sum is the probability of the new information, P ( B ). The sum .56 + .03 shows an overall probability of .59 of the drug working on human beings.

Bayes’ Theorem: Tabular Approach: 

Bayes’ Theorem: Tabular Approach (1) (2) (3) (4) (5) Events A i Prior Probabilities P ( A i ) Conditional Probabilities P ( B | A i ) A 1 A 2 .7 .3 1.0 .8 .1 .56 .03 Joint Probabilities P ( A i I B ) P ( B ) = .59

Bayes’ Theorem: Tabular Approach: 

Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability. The joint probabilities P ( A i I B ) are in column 4 and the probability P ( B ) is the sum of column 4. Bayes’ Theorem: Tabular Approach

Bayes’ Theorem: Tabular Approach: 

(1) (2) (3) (4) (5) Events A i Prior Probabilities P ( A i ) Conditional Probabilities P ( B | A i ) A 1 A 2 .7 .3 1.0 .8 .1 .56 .03 Joint Probabilities P ( A i I B ) P ( B ) = .59 Bayes’ Theorem: Tabular Approach .56/.59 Posterior Probabilities P ( A i | B ) . 9492 .0508 1.0000

End of Chapter 4: 

End of Chapter 4