ENGR 200 What did we do last week? Definition of probability xioms of probability Sample space robability laws Conditional probability ENGR 200 Lecture 3: genda. Conditional probability 2. Multiplication rule 3. The law of total probability 4. Bayes theorem
Definition of Conditional robability, B : two events ( B) : probability that event occurs given event B has already occurred ( B) ( B) (B) Conditional robability in ictures Before taking the test Random person comes to doctor Has Does not have no no Sample Space 2
Conditional robability in ictures N : event of no V : event of +: event that the test shows - : event that the test shows no (N) = 0.99 (+ V) = 0.90 (+ N) = 0.0 Random person comes to doctor Has Does not have no no Conditional robability in ictures fter the test shows robability that the person really has the given that the test shows? Random person comes to doctor Has Does not have no no New Sample Space 3
Conditional robability in ictures N : event of no V : event of + : event that the test shows - : event that the test shows no (V +) =? fter the test shows Random person comes to doctor Has Does not have no no Multiplication Rule Restatement of conditional probability (B) = (B ) () roof? (B C) = (C B)(B ) () 4
5 Multiplication (Chain) Rule ( B) = ( B) (B) Complete multiplication rule: )... ( )... ( ) ( ) ( )... ( ) ( 2 2 3 2 2 n n n n i i
Multiplication Rule icture No hearts in 3 cards Three cards are drawn from an ordinary 52-card deck without replacement. t each step each one of the remaining cards are equally likely to be picked. Find the probability that none of these three cards is a heart. 6
Example: roduct failure Consider a semiconductor manufacturing process, where we assume the following probabilities for product failure for given levels of contamination in manufacturing: robability Level of of Failure Contamination robability 0.0 High 0.2 0.0 Medium 0.3 0.00 Low 0.5 robability that a product using one of these chips fails? robability of high contamination in the production process if a product fails? Example: roduct failure 7
The law of total probability Let, 2, k be mutually exclusive and exhaustive (i.e., i =S) and B be an arbitrary event in S. Then the total probability of B is given by: roof... ( B) k i ( B ) ( i i ) The Chess Tournament You enter a chess tournament where your probability of winning a game is 0.3 against half the players, 0.4 against a quarter of the players, and 0.5 against the remaining players. What is the probability of winning against a randomly chosen opponent? 8
Biased coins Blue coin: (heads) = 0.99 Red coin: (heads) = 0.0 ick a coin at random, what is the probability of heads? Example: Modelling Radar Detection If an aircraft is present in a certain area, a radar correctly registers its presence with probability 0.99. If it is not present, the radar falsely registers an aircraft presence with probability 0.. We assume that an aircraft is present with probability 0.05. What is the probability that radar registers an aircraft? 9
Example: Radar Detection ={an aircraft is present} B={the radar registers an aircraft presence} ()=0.05 ( )=0.95 Example: Modelling Radar Detection If an aircraft is present in a certain area, a radar correctly registers its presence with probability 0.99. If it is not present, the radar falsely registers an aircraft presence with probability 0.. We assume that an aircraft is present with probability 0.05. What is the probability of an aircraft given that radar registered one? 0
Bayes theorem Let, 2, k be mutually exclusive and exhaustive (i.e., i =S) and B be an arbitrary event in S. Then for j=,,k: B ( j B) j j j B j,2,..., k k ( B) i B i i Example: Modelling Radar Detection If an aircraft is present in a certain area, a radar correctly registers its presence with probability 0.99. If it is not present, the radar falsely registers an aircraft presence with probability 0.. We assume that an aircraft is present with probability 0.05. What is the probability of an aircraft given that radar registered one?
Example: Radar Detection ={an aircraft is present} B={the radar registers an aircraft presence} ()=0.05 ( )=0.95 Biased coins Blue coin: (heads) = 0.99 Red coin: (heads) = 0.0 ick a coin at random, what is the probability of heads? If you know I got heads, what is the probability that I chose the blue coin? 2