Conditional ProbabilityConditional Probability, con’tSlide 3Slide 4Example-using definitionExample-Using Venn DiagramExampleExample, Using TableSlide 9Example, con’tSample Space for Rolling Two DieSlide 12Example-Tree DiagramIndependent EventsSlide 15Slide 16Independent Events, con’tConditional Probability and IndependenceIndependenceConditional ProbabilityP(A) represents the probability assigned to A, it is the original or unconditional probabilitySometimes there may be conditions “an event B has occurred” that affects the probability assigned to AThe conditional probability of an event, A, given that an event B has happened is denoted P(AB)P(AB) is read “the probability that A occurs given that B has occurred”Conditional Probability, con’tGiven that B has occurred, the relevant sample space has changed, it is no longer S but consists only of the outcomes in BA has occurred if and only if one of the outcomes in their intersection have occurredFor any events A and B with P(B)>0, ( )( )( )P A BP A BP BConditional Probability, con’tSAB( )( )( )P A BP A BP BConditional Probability, con’tWe can solve these problems using several methods:Use the formulaVenn DiagramsFrequency TablesTree DiagramsExample-using definitionThe 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(AB)P(BA)Example-Using Venn DiagramSuppose that A and B are events with probabilities: P(A)=1/3, P(B)=1/4, P(AB)=1/10Find each of the following using a Venn Diagram:1. P(AB)2. P(BA)3. P(ACB)4. P(BCA)5. P(ACBC)ExampleConsider the experiment of rolling a fair die twiceAll outcomes in S are equally likely 1,1 2,1 3,1 4,1 5,1 6,11, 2 2, 2 3, 2 4,2 5, 2 6, 21,3 2,3 3,3 4,3 5,3 6,31, 4 2, 4 3, 4 4,4 5, 4 6, 41,5 2,5 3,5 4,5 5,5 6,51, 6 2,6 3, 6 4,6 5,6 6, 6S Example, Using TableLet E=the sum of the faces is evenLet S2=the second die is a 2Find 1. P(S2E) 2. P(ES2) 1,1 2,1 3,1 4,1 5,1 6,11, 2 2,2 3, 2 4, 2 5, 2 6, 21,3 2,3 3,3 4,3 5,3 6,31, 4 2,4 3, 4 4, 4 5, 4 6, 41,5 2,5 3,5 4,5 5,5 6,51,6 2,6 3,6 4, 6 5, 6 6,6S Example, Using TableOne way of doing this is to construct a table of frequencies:Event AEvent AcTOTAL SEvent B Total BEvent BcTotal BcTotal ATotal AcGrand TotalA BCA BCA BC CA BExample, con’tLet’s try it to find P(S2E) and P(ES2)Event EEvent EcTOTAL SEvent S2Event S2c# of successes( )Total # of possible outcomesP E Remember:Sample Space for Rolling Two Die 1,1 2,1 3,1 4,1 5,1 6,11, 2 2, 2 3, 2 4, 2 5, 2 6, 21,3 2,3 3,3 4,3 5,3 6,31, 4 2, 4 3, 4 4, 4 5, 4 6, 41,5 2,5 3,5 4,5 5,5 6,51,6 2,6 3, 6 4,6 5,6 6,6S ExampleIf a fair coin is flipped three times, what is the probability that it comes up tails at least once given:1. No information at all2. All three coins produce the same side3. It comes up tails at most once4. The third flip is headsExample-Tree DiagramThree 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 a randomly chosen shock absorber installed by the manufacturer will be defective?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 EventsIf 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(EF)=P(E)P(F)Similarly, P(EF G)=P(E) P(F) P(G), etcIndependence of E and F implies that P(EF)=P(E) and P(F)= P(FE)If the events are not independent, then they are dependent.Independent EventsConsider 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 EventsYou 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? ExplainIndependent Events, con’tYou 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 5B: the red die shows a 22. A: the sum is 5B: the sum is less than 43. A: the sum is evenB: the red die is evenConditional Probability and IndependenceIf E, F and G are three events, then E and F are independent, given that G has happened, ifP(EFG)=P(EG) P(FG)Likewise, events E, F and G are independent, given that H has happened, given that G has happened, if P(EFGH)=P(EH) P(FH) P(GH)IndependenceIn 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
View Full Document
Unlocking...