 P(A c ∩ B) + P(A ∩ B)=P(B)  P(A c ∩ B) and P(A ∩ B) are called joint probability.  P(A) and P(B) are called marginal probability.  P(A|B) and P(B|A) are called conditional probability.

 Someone is shooting at a target. If it is windy, he has a 40% chance hitting the target. If there is no wind, his chance is 70%.  A. If there is a 12% chance of being windy, what is his chance of hitting the target?  B. If he hits the target, what is the chance of it being windy???

 Bayes theorem: ◦ Bayes theorem deals with another type of question. ◦ Think about conditional probability: it answers the question: if A happens then what is the chance for B to happen? ( A is a condition for B) ◦ Bayes theorem answers another question: If B happens, what is the chance of A happening. (A is still a condition for B )

 Think about total probability formula (two events case)  P(A)=P(B1)P(A|B1)+P(B2)P(A|B2) ◦ Now we want to know P(B1|A)

◦ P(B 1 |A) = P(B 1 )P(A|B 1 ) / [P(B 1 )P(A|B 1 )+P(B 2 )P(A|B 2 )] = P(B 1 )P(A|B 1 ) / P(A) ◦ P(B 2 |A) = P(B 2 )P(A|B 2 ) / [P(B 1 )P(A|B 1 )+P(B 2 )P(A|B 2 )] = P(B 2 )P(A|B 2 ) / P(A)

 It deals with the question that if we observe an outcome from an event, A, that is conditional on the outcome of another event, B, what is the probability for each of the outcomes of event B.  Compare with conditional probability: ◦ Conditional probability deals with that given the outcome of B, what is the probability of A.

 The denominator is always the probability that an outcome of A, which is found from total probability formula.  The numerator is the part in the total probability formula that addresses the outcome of B that we are interested in.

 All the problems start with two events, A and B. One event is always conditional on the other.  1. Determine which event is conditional on the other. If the outcome of A depends on the outcome of B, A is conditional on B.  2. Find out all the possible outcomes of B and their corresponding probabilities.  3. Find out which outcome of A actually occurred.  4. Construct the total probability formula.  5. Apply the Bayes’ formula.

 Let A={hit the target}, B={it is windy}. Then B C ={it is not windy}.  We want to calculate P(B|A)  By Bayes Theorem:  P(B|A)= P(B)P(A|B) / [P(B)P(A|B)+P(B C )P(A| B C )]  = P(B)P(A|B) / P(A)  = 0.12*0.4/(0.12* *0.7  =0.07

 In an exam, there is a problem that 60% of students know the correct answer. However, there is 15% chance that a student picked the wrong answer even if he/she knows it and there is also a 25% chance that a student does not know the answer but guessed it correctly. If a student did get the problem right, what is the chance that this student really knows the answer? What if he/she did not get it right?

 In order to detect whether a suspect is lying, police sometimes use polygraph. Let A={polygraph indicates lying} and B={the suspect is lying}. If the suspect is lying, there is a 88% chance of detecting it; if the suspect is telling the truth, 86% of time the polygraph will confirm it. We assume that 1% of the time the suspects lie. If the result of polygraph shows that the suspect is lying, what is the chance that this person is really lying?

 There is a new disease that the authority believe 30% people are infected. A company provided a test that can detect whether a person has the disease or not. If the person really has the disease, the test will miss it 10% of the time. If the person is not infected, the test will show negative 75% of the time. If someone has got a positive result, what is his/her chance of really having the disease?

 Always have a good idea of which outcome are we looking at, so that we can correctly set up the total probability formula.

 Suppose you want to catch an early flight at Indy airport. You have the following plans: ◦ 1. Let your friend drive you to the airport, with 20% chance of missing the flight. ◦ 2. Taking the Lafayette Limo with a 25% chance of missing the flight. ◦ 3. Hitch hike yourself with 50% of missing the flight. If your preference is 40% taking Lafayette Limo, 35% hitch hike and 25% asking your friend to drive, what is your chance of missing the flight.  If you actually caught the flight, what is your chance of choosing hitch hike?

 Kokomo, Indiana. In Kokomo, IN, 65% are conservatives, 20% are liberals and 15% are independents.  Records show that in a particular election 82% of conservatives voted, 65% of liberals voted and 50% of independents voted.  If the person from the city is selected at random and it is learned that he/she did not vote, what is the probability that the person is liberal?