A. 1 Elementary Probability Set Theory Experiment -- the process by which an observation is made Ex. Outcome The result of a chance experiment. Ex. Sample Space -- ( S) the collection of ALL possible outcomes of an experiment Ex. The sample space for flipping 2 coins: Event --A special subset of interest Notation: A, B, C,... Ex. List the event where you get the same face on both flips. Empty Event An event that has no outcomes ( )
Type of Event Symbols Definition Venn Diagram Union All points in A or B or both Intersection All points in A and B simultaneously Complement All points not in A Type of Event Definition Venn Diagram Mutually Exclusive Two events that have no outcomes in common Not mutually exclusive Two events that have something in common
A.1.2 Simple Theorems of Probability Theory The probability of an event (or outcome) is the proportion of times the event would occur in a long run of repeated experiments. A) the _ number _ of _ outcomes _ of _ an _ event the _ number _ of _ outcomes _ in _ S Rules of Probability Probability is a number between 0 and 1. P (S). empty set). If A 1, A 2,... is a collection of mutually exclusive events, then o P ( A 1 ora 2 ora...) P ( A ) ( ) ( )... 3 1 + P A 2 + P A 3 + For any event A, not) 1- P (A). If A and B are mutually exclusive then AandB) 0. Why? For any two events A and B, A B ) A) + B ) A B).Why? Ex. The probability that an integrated circuit chip will have defective etching is 0.12, the probability that it will have a crack defect is 0.29, and the probability that it has both defects is 0.07. a. What is the probability that a newly manufactured chip will have either an etching or a crack defect or both? b. What is the probability that a newly manufactured chip will have neither defect?
A.1.3 More Advanced Probabilities The conditional probability of event A given B, is the probability that event A occurs given that event B occurs. A B) A B) B ) The multiplicative rule is simply the conditional probability rule rewritten. B A) A B) P ( B) Independence is when knowing something about A tells you nothing about B. P (A B) P (A) Examples 1. Show that P ( A B) P ( A) * B). 2. A certain system can experience three different types of defects. Let A i (i 1,2,3) denote the event that the system has a defect of type I. Suppose that A 1 )0.12 A 2 )0.07 A 3 ) 0.05 A 1 A 2 ) 0.06 A1 A2 A3 ) 0.01 a. What is the probability that the system does not have a type I defect? b. Given that the system has a type 1 error, what is the probability that it has a type 2 defect?
3. The probability that a microchip fails on its first use is 0.10. Given that a microchip lasts through its first use, the probability that it lasts a year is 0.99. What is the probability that the chip doesn't fail during its first year (including first use)? 4. A rocket has a built-in redundant system. In this system, if component K 1 fails, it is bypassed and component K 2 is used. The probability of failure of any one of these components is 0.15. Assume that the failures of these components are mutually independent events. What is the probability that the system does fail? Extra Problems 1.Thirty cameras are in a shipment. Unknown to the buyer, 6 are defective. Buyer chooses two at random and tests them. Accept shipment only if both cameras are nondefective. Find the probability that they accept the shipment. 2.There are 6 field joints in the Solid Rocket Booster for the Space Shuttle. If one of the joints fails, then there is disaster. At 60 degrees F, the probability, that each of the joints works, is.9968. Assume that the joints are independent. What is the probability that all six field joints work?
3. Survey of High School students in Arizona asked for smoking habits of kids and parents. Parents Students Smokes Student does not smoke Totals Both smoke 400 1380 One smokes 416 1823 Neither smokes 188 1168 Totals a.) Find the probability of a student being a smoker given that both parents smoke b.) Find the probability that both parents smoke given that the student smokes c.) Are being a smoker and having both parents a smoker independent events?