For Thursday, Feb. 20

Bring Conditional Probability Worksheet to classExam #1: March 4Project Presentations Start March 6Team Homework #4 due Feb. 25READ: Project 1: Baye’s Theorem

FOCUS LESSON

Conditional Probability

P(A) represents the probability assigned to A -- it is the “unconditional” probabilitySometimes there may be conditions that affect the probability assigned to A; “an event B has occurred” The conditional probability of an event, A, given that an event B has happened is denoted P(A | B)P(A | B) is read “the probability that A occurs given that B has occurred”

Conditional Probability, con’t

Given that B has occurred, the relevant sample space has changed; it is no longer S but consists only of the outcomes in BFor any events A and B with P(B)>0,

( )( )

( )

P A BP A B

P B

Conditional Probability, con’t

S

A B

( )( )

( )

P A BP A B

P B

Conditional Probability, con’t

We can solve these problems using several methods:Use the formulaVenn DiagramsFrequency TablesTree Diagrams

Example-Using definition

The probability that event A occurs is .63. The probability that event B occurs is .45. The probability that both events occur is .10. Find by using the definition:P(A | B)P(B | A)

Exercise #1

Suppose that A and B are events with probabilities: P(A)=1/3, P(B)=1/4, P(A ∩ B)=1/10Find each of the following using a Venn Diagram:1. P(A | B)2. P(B | A)3. P(AC | B)4. P(BC | A)5. P(AC | BC)

Example

Consider the experiment of rolling a fair die twiceAll outcomes in S are equally likely

1,1 2,1 3,1 4,1 5,1 6,1

1,2 2,2 3,2 4,2 5,2 6,2

1,3 2,3 3,3 4,3 5,3 6,3

1,4 2,4 3,4 4,4 5,4 6,4

1,5 2,5 3,5 4,5 5,5 6,5

1,6 2,6 3,6 4,6 5,6 6,6

S

Example, Using Table

Let E=the sum of the faces is evenLet S2=the second die is a 2Find 1. P(S2 | E) 2. P(E | S2)

1,1 2,1 3,1 4,1 5,1 6,1

1,2 2,2 3,2 4,2 5,2 6,2

1,3 2,3 3,3 4,3 5,3 6,3

1,4 2,4 3,4 4,4 5,4 6,4

1,5 2,5 3,5 4,5 5,5 6,5

1,6 2,6 3,6 4,6 5,6 6,6

S

Example, Using Table

One way of doing this is to construct a table of frequencies:

Event A Event Ac TOTALS

Event B Total B

Event Bc Total Bc

Total A Total Ac Grand Total

A B CA BCA B C CA B

Example, con’t

Let’s try it to find P(S2|E) and P(E|S2)

Event E Event Ec TOTALS

Event S2

Event S2c

# of successes( )

Total # of possible outcomesP E Remember:

Example

Let S be the event that a person is a smoker and let D be the even that a person has a disease. The probability that a person has a disease is .47 and the probability that a person is a smoker is .64. The probability that a smoker has a disease is .39.– Find the probability that a person with a disease is a

smoker

Example-Tree Diagram

Three manufacturing plants A, B, and C supply 20%, 30% and 50%, respectively, of all shock absorbers used by a certain automobile manufacturer. Records show that the percentage of defective items produced by A, B and C is 3%, 2% and 1%, respectively. – What is the probability that the part came from manufacturer

A, given that the part was defective?– What is the probability that the part came from B, given that

the part was not defective?

Independent Events

If two events are independent, the occurrence of one event has no effect on the probability of the other.E and F are independent events if P(E ∩ F)=P(E) * P(F)Similarly, P(E ∩ F ∩ G)=P(E)* P(F) * P(G), etcIndependence of E and F implies that P(E | F)=P(E) and P(F)= P(F | E)If the events are not independent, then they are dependent.

Independent Events

Consider flipping a coin and recording the outcome each time.Are these events independent?Let Hn=the event that a head comes up on the nth tossWhat is the P(H1 | H2)?

What is the probability P(H1 ∩ H2 ∩ H3)?

Independent Events

You throw two fair die, one is green and the other is red, and observe the outcomes.Let A be the event that their sum is 7Let B be the event that the red die shows an even #

Are these events independent? Explain.Are these events mutually exclusive? Explain

Independent Events, con’t

You throw two fair die, one green and one red and observe the numbers. Decide which pairs of events, A and B, are independent:1. A: the sum is 5

B: the red die shows a 22. A: the sum is 5

B: the sum is less than 43. A: the sum is even

B: the red die is even

Conditional Probability and Independence

If E, F and G are three events, then E and F are independent, given that G has happened, ifP(E ∩ F | G)=P(E | G) *P(F | G)Likewise, events E, F and G are independent, given that H has happened, given that G has happened, if P(E ∩ F ∩ G | H)=P(E | H) * P(F | H) * P(G | H)

Independence

In the manufacture of light bulbs, filaments, glass casings and bases are manufactured separately and then assembled into the final product. Past records show: 2% of filaments are defective, 3% of casings are defective and 1% of bases are defective.What is the probability that one bulb chosen at random will have no defects?

Focus on The Project

So, we have found the probability of success and the probability of failure, based on all the recordsUsing information about our borrower might change our probability of success and failure

Focus on the Project

Let S and F be the events of success and failure, respectivelyLet Y be the event of having 7 years of experienceLet T be the event of having a Bachelor’s DegreeLet C be the event of having a normal state of economy

Focus on the Project

In terms of conditional probability, we would like to know P(S | Y), P(S | T), and P(S | C)We would also like to know the corresponding probabilities for failureWe can estimate these from our bank recordsWhen we find P(S | Y) we are implying we are looking at BR bank, etc.

Focus on the Project

How can we find P(S | Y)?Once we have this value, we can find the other conditional probabilities in the same wayWith the conditional probabilities, we can find the correlating expected valuesBased on these expected values, we can revise our decision on whether to foreclose or workout?

Focus on the Project

What potential problem do we encounter when we look at the expected values that we just found?We would like to find P(S | Y T C) and

P(F | Y T C) -- unfortunately our bank records do not hold this information so we can’t find it directlySo, let’s calculate something that we can find AND will be of importance to us later in finding out the above probabilities

Focus on the Project

Let’s find P(Y T C | S) and then in a similar fashion, P(Y T C | F)The project description says that Y, T, and C are independent events, even when they are conditioned upon S or F.Hence, P(Y T C | S) = P(Y|S)*P(T|S)*P(C|S)P(Y T C | F) is similar

What do you need to do?

Calculate P(Y T C | S) and P(Y T C | F) using your borrower’s

informationMake sure you are keeping all of your information in an Excel fileOnce we have these numbers, we are going to learn how to use these numbers to find out what we need to know (but can’t get directly)