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Ch. 11 Conditional Probability • P(A|B) = P(A and B)/ P(B), assuming P(B) > 0 • P(B|A) = P(A and B)/ P(A), P(A) > 0 Examples • In a monthly report, the local animal shelter states that it currently has 24 dogs and 18 cats available fop adoption. Eight of the dogs and six of the cats are male. Find each of the following probabilities. • a) The pet is male given that it is a cat • b) The pet is a cat, given that it is female • c) The pet is dog given that it is a female Example • You draw a card at random from a standard deck of 52 cards. Find the following probabilities. • a) The card is a heart, given that it is red • b) The card is red, given that it is a heart • c) The card is an ace, given that it is red • d) The card is a queen, given that it is face card Ch. 11 Conditional Probability Ch. 14/15 From randomness to Probability Example • 3. The Masterfoods Company says that before the introduction of purple, yellow candies made up of 20% of their plain M&M’s, red another 20% and orange, blue and green each made up 10%. The rest were brown. • • • • a) If you pick an M&M at random, what’s the probability that i) it is brown ii) it is yellow or orange? iii) it is not green? Examples • 4. If you pick three M&M’s in a row, what s the probability that • a) they are all brown? • b) the third one is the first one that is red? • c) none are yellow? • d) at least one is green Example • The probabilities that an adult American man has high blood pressure and/or high cholesterol are shown in the table High Cholesterol OK HIGH Blood Pressure OK 0.11 0.21 0.16 0.52 Example • a) What is the probability that a man has both conditions? • b) What is the probability that a man has high blood pressure? • c) What is the probability that a man with high blood pressure has high cholesterol? • d) What is the probability that a man has high blood pressure if it is known he has high cholesterol? • e) Are high blood pressure and high cholesterol independent? Example • A poll conducted by the D University of Montana classified respondents by gender and political Male 36 party as shown in the table. Is party affiliation independent of gender? Female 48 R I 45 24 33 16 Example D R I M .18 .22 .12 .52 F .24 .16 .08 .48 .42 .38 .20 1 Chapter 11 Conditional Probability and Tree Diagram 1% of employees in a company are drug users. A test was developed to identify the drug users. If a person is a drug user, then the test returns positive result 98% of the time. If a person is not a drug user, the test returns negative result 97% of the time. Will you recommend this test to be administered at your company? Conditional Probability and Tree Diagrams Leah is flying from Boston to Denver with a connection in Chicago. The probability her first flight leaves on time is 0.15. If the flight is on time, the probability her luggage will make the connection flight in Chicago is 0.95, but if the first flight is delayed, the probability that the luggage will make it is only 0.65. a) What is the probability that her luggage arrives on time? Answer: 0.695 b) Are the first flight leaving on time and the luggage making the connection independent events? NO Conditional Probability and Tree Diagrams A package will be picked up by one of 3 delivery vans, depending on which is nearest. The choices are equal for any of the vans to make the pick up. Past record for each van’s success in delivering packages are given as follows: • Prob(on time delivery for van 1) = .95 • Prob(on time delivery for van 2) =.96 • Prob(on time delivery for van 3) = .90. • What is the probability a package will be delivered on time? ANS 0.94 • Is package deliver independent of delivery van? NO • If package is delivered on time, what is the probability that it was delivered by Van 1? ANS 0.338 Bayes’ Theorem-Reversing the Conditioning P ( B | A) P( A | B)P(B) c c P( A | B)P(B) P( A | B )P(B ) At a gas station, 40% of customers pump regular gas, 35% pump midgrade gas, and 25% pump premium grade gas. Of those who pump regular, 30% pay at least $30. Of those who pump midgrade, 50% pay at least $30. And of those who pump premium, 60% pay at least $30. a) What is the probability that the next customer will pay more than $30? ANS 0.445 b) If the next customer pays more than $30, what is the probability that he pumped regular gas? ANS 0.269 Reversing the Condition A firm that specializes in filing out income tax forms has two types of accounts • Large(gross income greater than $50000) • Small(gross income less than $50000) They find that 5% of the large accounts and 2% of the small accounts are audited by the IRS. If 40% of their accounts are large accounts, and an account is audited, what is the probability that it is large? ANS 0.625