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Chapter 6 Review Fr Chris Thiel 13 Dec 2004 What is true about probability? • The probability of any event must be a number between 0 and 1 inclusive • The sum of all the probabilities of all outcomes in the sample space must be 1 • The probability of an event is the sum of the outcomes in the sample space which make up the event Independent Iff P(B | A) P(B) Previous outcomes do not change probability Multiplication Rule: P(A and B)=P(A)P(B) Disjoint One outcome precludes the other since there is No overlap… Complement A A notA A c The event A does not occur P(A ) 1 P(A) c Addition Rules P(A or B)=P(A)+P(B)-P(A and B) Multiplication Rules P(A and B)=P(A)P(B) if A and B are independent Conditional Rules P(A B) P(A)P(B | A) P(A B) P(B | A) P(A) P(65+)=18% P(Widowed)=10% a. If among 65+, 44% widowed, What percent of the population are widows over 65? P(65 W ) P(65)P(W | 65) (.18)(.44) .0792 b. If 8% are widows over 65, What is the chance of being a widow given that they’re over 65? P(W 65) P(W | 65) P(65) .08 .44 .18 See Table 6.1 p. 366 Use Venn Diagrams & Trees Venn Diagrams can help see if events are Independent, complementary or disjoint Use Tree Diagrams to Organize addition and Multiplication rules to combinations of events If event A and B are disjoint, then • P(A and B)= 0 • P(A or B) =1 • P(B)=1-P(A) Independent events… you flip a coin and it’s heads 4 times in a row…. The odds are STILL the same The 6 is 3 times more likely to occur… what is the probability of rolling a 1 or a 6? x x x x x 3x 1 3 8 18 1 2 A fair die is tossed 4 or 5-win $1 6-win $4 If you play twice: what is the probability that you will win $8? $2? P(A)=.5 P(B)=.6 P(A andB)=.1 • • • • • A B .4 .1 .5 P(A|B)=? Are A and B Independent? Disjoint? Will either A or B always occur? Are A and B complementary? 0 Lie Detector • Reports “Lie” 10% if person is telling the truth • Reports “Lie” 95% if the person is actually lying • Probability of machine never reporting a lie if 5 truth tellers use it (.9) .59049 5 You enter a lottery, the odds of getting a prize is .11 If you try 5 times, what is the probability that you will win at least once? • 1-P(never winning) 1 (.89) 5 8% have a disease. A test detects the disease 96% And falsely indicates the disease 7%. If you test positive, what is the chance you have the disease? Tests + .96 (.08)(.96)=.0768 .04 Tests - (.08)(.04)=.0032 Tests + .07 (.92)(.07)=.0644 .93 (.92)(.93)=.85 Has Disease .08 .92 No Disease Tests - P(D|+) P(D ) P(D | ) P() (.0768) .65 (.0768) (.0644) P(Harvard)=40% P(Florida)=50% P(both)=20% P(none)=? P(F but not H)=? H F .2 .2 .3 .3 30% of calls result in a airline reservation. a. P(10 calls w/o a reservation)=? (1 .3) .0282 10 b. P(at least 1 out of 10 calls has a reservation)=? 1 P(none) 1 .0282 .9718 85% fire calls are for medical emergencies Assuming independence… P(exactly one of two calls is for a medical emergency)=? P(M)P(F)+P(F)P(M)=(.85)(.15)+(.15)(.85)=.255 Is it really independent?