Lecture 7: Complex Events and Conditional Probabilities Chapter 5: Probability and Sampling Distributions 1/30/12 Lecture 7 1
5.1 Chance Experiments A chance experiment (or a random experiment) - An activity or situation whose outcomes depend on chance to some degree. - To determine whether a given activity qualifies as a chance experiment, ask: Will I get exactly the same result if the experiment is repeated more than once? If the answer is No, then it is a chance experiment. - Examples: - determine whether a metal part withstands a stress test, - record if it rains tomorrow, - measure the yield of a chemical reaction, - assess the potency of a pharmaceutical product, - measure the volume of water flow in a drainage system.
Events To compute probabilities, we need to work with the outcomes of chance experiments, which can be divided Into two types: Simple Events: individual outcomes Events: collections of simple events Sample Space: Obtained by taking all simple events together Venn Diagrams useful for depicting relationships between events
Venn Diagrams See from Pg 195, the textbook
Example Consider: An experiment of tossing a coin three times Simple Events: e.g. HHT, THT,. How many in total? 8 simple events: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT What is the sample space? {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} What is event A that we get at least two heads? A={HHT, HTH, THH, HHH}
Example 5.1 Conducting a series of stress tests on four metal parts What is the sample space? Event A: at least two parts pass the stress test Event B: at most two parts pass the stress test Event C: exactly three parts pass the test How to express all the above 4 events? (Sample space is also one kind of event) 1/30/12 Lecture 7 6
Sample Space: Continued S = {PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF}, by the tree diagram. Event A: at least two parts pass the stress test, so A={PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP} Event B: at most two parts pass the stress test, thus B={PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF} Event C: exactly three parts pass the test, therefore 1/30/12 C={PPPF, PPFP, Lecture 7 PFPP, FPPP} 7
A Tree Diagram See it from Fig. 5.2 on Pg 194 1/30/12 Lecture 7 8
Complex Events Let A and B be any two events of a chance experiment Event A or B: consists of all simple events that are contained in either A or B. i.e. At least one of A or B occur. Event A and B: consists of all simple events that A and B have in common. i.e. Both A and B occur. Event A (Complement of A): consists of all simple events that are NOT contained in A Disjoint Events (Mutually Exclusive): two events that have no simple events in common 1/30/12 Self-reading for generalized Lecture 7 definition (Pg 197) 9
Still, Example 5.1 Review: Conducting a series of stress tests on four metal parts 1. Event A: at least two parts pass the stress test A={PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP} 2. Event B: at most two parts pass the stress test B={PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF} What are: Event A and B, Event A or B, and Event A, respectively? A Case of Disjoint Events: Event A and Event A Based on definition of A complement 1/30/12 Lecture 7 10
Probability Concepts Probability Axioms: 0 P(A) 1 for any event A P(S) = 1, where S is the sample space To Determine Probabilities? Frequencies of occurrence From subjective estimates Assume simple events are equally likely Use density and mass functions Note: Depending on the circumstances, each method has its merits. 1/30/12 Lecture 7 11
Addition Rules Addition Rule - For any two disjoint events A and B, P(A or B) = P(A)+P(B) Complementary Events: P(A ) = 1 - P(A) General Addition Rule: (for any two events A and B) P(A or B) = P(A)+P(B)-P(A and B) 1/30/12 Lecture 7 12
Example 5.1 Conducting a series of stress tests on four metal parts Event A: at least two parts pass the stress test A={PPPP, PPPF, PPFP, PFPP, FPPP, PPFF, PFPF, PFFP, FPPF, FPFP, FFPP} Event B: at most two parts pass the stress test B={PPFF, PFPF, PFFP, FPPF, FPFP, FFPP, PFFF, FPFF, FFPF, FFFP, FFFF} P(A and B) =? P(A or B) =? P( A ) =? 1/30/12 Lecture 7 13
Conditional Probability Let A and B are two events with P(B)>0. The conditional probability of A occurring given that event B has already occurred is denoted by P(A B), where P(A B) = P (A and B) / P(B) 1/30/12 Lecture 7 14
Independent Events Two events, A and B, are independent events if the probability that either one occurs is not affected by the occurrence of the other. In this case, P (A and B) = P(A) P(B) Self-Reading: Air Filter on Pg 205, bottom; Independence for more than 2 events: Pg 206 If A and B are independent, so are A and B. Question: If two events, A and B, are independent, what is P(A B)? 1/30/12 Lecture 7 15
Example 5.6 Suppose that the switches A and B in a twocomponent series system are closed about 60% and 80% of the time, respectively. If we assume that the closing of switch A occurs independently of switch B, what is the probability that the entire circuit is closed? Suppose that A and B are independent events with P (A)=0.5 and P(B) = 0.4. Can A and B be mutually exclusive? 1/30/12 Lecture 7 16
After Class Review Sec 5.1 through 5.3 Start Hw#3 Lab#2 will be held this Wed Hw#2 is due today, 5pm. Office hour Dropbox beside my officedoor 1/30/12 Lecture 7 17