University of California, Los AngelesDepartment of StatisticsStatistics 13 Instructor: Nicolas ChristouHomework 3EXERCISE 1You draw a card at random from a standard deck of 52 cards. Find each of the following conditionalprobabilities.a. The card is a heart, given that it is black.b. The card is black, given that it is a heart.c. The card is an ace, given that it is black.d. The card is a queen, given that it is a face card.EXERCISE 2The probabilities that an adult American man has high blood pressure and/or high cholesterol areshown in table below:Blood PressureHigh OKHigh 0.11 0.21Cholesterol OK 0.16 0.52An adult American is randomly selected:a. What is the probability that he has both conditions?b. What is the probability that he has high blood pressure?c. Suppose that this a man has high blood pressure. What is the probability that he has highcholesterol?d. What is the probability that this man has high blood pressure if it is known that he has highcholesterol?EXERCISE 3Use the table of the previous exercise: Are high blood pressure and high cholesterol independent?EXERCISE 4Suppose a woman tries on a dress. The probability that she asks for alterations is 0.65. Theprobability that she asks for delivery is 0.32. The probability that she asks for alterations anddelivery is 0.21. Suppose that you randomly select a woman who has tried on a dress. What is theprobability that:a. She will either ask for alterations or for delivery of the dress or both.b. She will not ask for alterations and she will not ask for delivery of the dress.EXERCISE 5A tennis player A has probability of23of winning a set against player B. A match is won by theplayer who first wins three sets. Find the probability that A wins the match.EXERCISE 6Observations of a waiting line at a medical clinic indicates that the probability that a new arrivalwill be an emergency case is p =16. Find the probability that the rthpatient is the first emergencycase. Assume that conditions of arriving patients represent independent events.EXERCISE 7Given that a person has a certain disease, a diagnostic test will detect it with probability 0.90.Also, given that a person does not have the disease, the diagnostic test will detect that the personhas the disease with probability 0.10. Only 1% of the population has the disease in question.You can use:T= {diagnostic test detects that a person has the disease}D= {person actually has the disease}a. A person is randomly selected from the population and tested for this disease. What is theprobability that the diagnostic test will detect that the person has the disease?b. A person is chosen at random from the population. Given that the diagnostic test detectsthat the person has the disease, what is the probability that the person actually has thedisease?EXERCISE 8Employment data at a large company reveal that 72% of the workers are married, that 44% arecollege graduates, and that half of the grads are married. What is the probability that a randomlychosen workera. is neither married nor a college graduate?b. is married but not a college graduate?c. is married or a college graduate?EXERCISE 9Suppose that 23% of adults smoke cigarettes. It is known that 57% of smokers and 13% of non-smokers develop a certain lung condition by age 60.a. Explain how these statistics indicate that lung condition and smoking are not independent.b. What is the probability that a randomly selected 60-year-old has this lung condition?EXERCISE 10Three different machines M1, M2, and M3were used for producing a large batch of similar man-ufactured items. Suppose that 20% of the items were produced by machine M1, 30% by machineM2, and 50% by machine M3. Suppose further that 1% of the items produced by machine M1aredefective, that 2% of the items produced by machine M2are defecive, and that 3% of the itemsproduced by machine M3are defecive.a. Find the probability that an item selected at random will be found defective.b. Suppose that one item is selected at random from the entire batch, and it is found to bedefective. What is the probability that this item was produced by machine
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