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1 Review Probability Random variables Binomial distribution

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2 1. Event A occurs with probability 0.2. Event B occurs with probability 0.8. If A and B are disjoint (mutually exclusive) then (i) p(A and B)=0.16 (ii) p(A or B)=1 (iii) p(A and B)=1 (iv) p(A or B)=0.16 2. Ignoring twins and other multiple births, assume babies born at a hospital are independent events with probability that a baby is a boy and that a baby is a girl both equal to 0.5. The probability that the next 5 babies are girls is: (i) 1 (ii) 2.5 (iii) 0.25 (iv) 0.03125 (v) 0.5

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3 3. In a certain town 50% of the households own a cellular phone, 40% own a pager and 20% own both a cellular phone and a pager. The proportion of households that own neither a cellular phone nor a pager is (i) 10% (ii) 30% (iii) 70% (iv) 90% 4. Event A occurs with probability 0.3 and event B occurs with probability 0.4. If A and B are independent, we may conclude that (i) p(A and B)=0.12 (ii) p(A|B)=0.3 (iii) p(B|A)=0.4 (iv) all of the above

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4 5. Of all children in a juvenile court, the probability of coming from a low income family was.60; the probability of coming from a broken home was 0.5; the probability of coming from a low-income broken home was 0.40 (i) what is the probability of coming from a low-income family or broken home (or both)? (iii) find the probability of coming from a broken home, given that it was a low income family. Are the two events low-income and broken home independent?

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5 6.Jane and Tim prepare their wedding invitations by themselves. Jane works faster and prepares 80% of the invitations. However 10% of her invitations turn out with some mistakes. Out of Tim’s invitations, only 1% have mistakes. (iii) What is the probability of an invitation has a mistake in it? (iv) Given that an invitation has a mistake, what is the probability that it has been written by jane?

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6 7. A certain university has the following probability distribution for number of courses X taken by seniors in their final semester courses 1 2 3 4 5 6 Probability.05.10.30.40.10.05 (i) What is the probability that a randomly chosen senior took at least 4 courses in the final semester? (ii) what is the probability that a randomly chosen senior took more than 4 courses in the final semester? (ii) The shape of the distribution of number of courses is: skewed left / skewed right / symmetric but not at all bell-shaped / reasonably bell shaped

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7 8. A surprise quiz contains 3 multiple choice questions. Question 1 has three suggested answers, Question 2 has three suggested answers, and question 3 has two. A completely unprepared student decides to choose the answers at random. Let X be the number of questions that the student answers correctly. a.List the possible values of X X=0,1,2,3 b.Find the probability distribution of X. Q1-first is correct p(Q1)=1/3 Q2-second is correct p(Q2)=1/3 Q3-third is correct p(Q3)=1/2

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8 9. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5 you win $1, if number of spots showing is 6 you win $4, and if the number of spots showing is 1,2, or 3 you win nothing. Let X be the amount that you win. The expected value of X (mean of X) is: i) $0.00 (ii) $1.00 (iii) $2.50 (iv) $4.00

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9 10. A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability: The mean number of customers that make a purchase during the first hour that the store is open is i) 2 (ii) 2.5 (iii) 3 (iv) 4 X01234 P(x)0.1 0.6

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10 11. 8% of males are color blind. A sample of 8 men is taken and the number X of people that are color blind are counted. What is the probability to find 4 people that are color blind in the sample? what is the probability that at least 7 people in the sample are color blind?

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