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1 probability

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Specify Sample Space 1-1: Toss a coin two times and note the sequence of heads and tails. 1-2: Toss a coin three times and note the number of heads. 1-3: Pick two real numbers at random between zero and one. 1-4: Pick a real number X at random between zero and one, then pick a number Y at random between zero and X.

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HW1-1 (to be posted) Three systems are depicted, each consisting of 3 unreliable components. The series system works if and only if (abbreviated as iff) all components work; the parallel system works iff at least one of the components works; and the 2- out-of-3 system works iff at least 2 out of 3 components work. Find the event that each system is functioning.

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Prove the theorems 1-5: P( ) = 0 1-6: A B => P(A) P(B) 1-7: P(A) 1 1-8: P(A c ) = 1 – P(A) 1-9: P(A B) = P(A) + P(B) – P(A B)

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1-10: Assign the probability Probability: The random experiment is to throw a fair die. How can we define sample space S, and probability law P to an arbitrary event E (that belongs to 2 S )?

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1-11: Find the probability

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1-12: Conditional Prob. An urn contains two black balls, numbered 1 and 2, and two white balls, numbered 3 and 4. S = {(1,b), (2,b), (3,w), (4,w)} A: black balls are selected, B: even-numbered balls are selected C: number of selected ball is greater than 2 Assuming that the four outcomes are equally likely, find P[A|B] and P[A|C]

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1-13: Bayes’ rule (1/2) Young actors are more drug-addictive among all actors? Sample space is young and old actors Drug-addicted young actors 1050 30 60 1020 70 90 old actors Y O D+D-

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1-13 Bayes’ rule (2/2) P[Y|D+] = P[Y D+] / P[D+] P[Y] = 30/90 P[Y D+] = P[D+|Y] * P[Y] = 10/30 * 30/90 P[D+] = P[D+|Y] * P[Y] + P[D+|O] * P[O] = 10/30 * 30/90 + 10/60 * 60/90 = 20/90 P[Y] = 30/90 P[Y|D+] = 10/90 / 20/90 = 1/2

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HW 1-2 (to be posted) Suppose a drug test is 99% true positive and 99% true negative results. Suppose that 0.5% of people are users of the drug. If a guy tests positive, what is the probability he is a real drug user?

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HW 1-3 (to be posted) Go back to young actor drug problem. Which values are to be in the blanks if we want to conclude that young actors are not more drug- addictive? Drug-addicted young actors ?? 30 60 1020 ?? 90 old actors Y O D+D-

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A ACAC BCBC B

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Simple Mathematical Facts for Lecture 1. Conditional Probabilities Given an event has occurred, the conditional probability that another event occurs.

Simple Mathematical Facts for Lecture 1. Conditional Probabilities Given an event has occurred, the conditional probability that another event occurs.

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