You will NOT earn any credits unless you show the intermediate steps to obtain your solution and the final solution is exactly correct. (1)(5 points) A popular breakfast hangout is located near a campus. A typical breakfast there consists of one beverage, one bowl of cereal,and a piece of fruit.If you can choose among three different beverages, seven different cereals and four different types of fruit, how many choices for breakfast do you have? (2)(5 points for each sub-problem) A bag contains 45 beans of three different varieties. Each variety is represented 15 times in the bag. You grab 9 beans out of the bag. (a) Count the number of ways that each variety can be represented exactly three times in your sample. (b) Count the number of ways that only one variety appears in your sample. (3)(6 points) Five cards are drawn at random without replacement from a standard deck of 52 cards. What is the probability of three of a kind and a pair (for instance,QQQ33)? (This is called a full house in poker.) (4)(8 points) An urn contains three red and two blue balls. You remove two balls without replacement. What is the probability that the two balls are of different colors? (5)(8 points) Suppose that you have a batch of red- and white-flowering pea plants, and suppose also that all three genotypes CC , Cc and cc are equally represented in the batch. (The red-flowering pea plant is with genotypes CC or Cc while the white-flowering pea plant is with genotype cc.) You pick one plant at random and cross it with a white-flowering pea plant. What is the probability that the offspring will have red flowers? (6)(8 points) An urn contains three blue and two white balls. You draw a ball at random, note its color, and replace it. You repeat these steps three times. Let X denote the total number of white balls. Find Prob(X≦1). (7)(5 points for each sub-problem) Suppose that the probability mass function of a discrete random variable X is given by the following table. ┌──┬───┐ │x │P(X=x)│ ├──┼───┤ │0 │0.3 │ ├──┼───┤ │1 │0.3 │ ├──┼───┤ │2 │0.1 │ ├──┼───┤ │3 │0.1 │ ├──┼───┤ │4 │0.2 │ └──┴───┘ (a) Find E[X]. (b) Find E[X^2]. (c) Find E[2X-1]. (8)(5 points for each sub-problem) A random variable X is said to have a geometric distribution if its probability mass function is given by Prob{X = n} = [(1-p)^(n-1)]p with n = 1,2,... and 0 < p < 1. If P = 1/3 , compute (a) E[X] and (b) Var[X]. (You may either use the formulae provided in our text book to obtain your solution or to derive the formulae by yourself.) (9)(10 points) A laboratory blood test is 95 percent effective in detecting a certain disease when it is, in fact, present. However, the test also yields a "false positive" result for 1 percent of the healthy persons tested. (That is, if a healthy person is tested, then, with probability .01, the test result will imply that he or she has the disease.) If 0.5 percent of the population actually has the diseases, what is the probability that a person has the disease given that the test result is positive? (10)(5 points for each sub-problem) Consider a discrete random variable Y with the following c.d.f. ╭ │0　　　　y < 1 　　　　　　　　　　　　　　│1/4　　1 ≦ y < 2 F(y)= │3/4　　2 ≦ y < 3 │7/8 3 ≦ y < 4 　　　　　　　　　　　　　　│1 4 ≦ y 　　　　　　　　　　　　　　╰ Find out the probability for (1) Prob{Y = 2} and (2) Prob{1 < Y ≦ 3 }. (11)(10 points) A discrete random variable X has the following probability mass function: p(x) ↑ 1/2 │ │ │ │ │ │ │ 1/3 │ │ │ │ │ │ │ 1/6 │ │ │ │ │ │ ───┴──┴──┴──┴───────→ -2 -1 0 1 What is its cumulative distribution function? (You may either give its precise mathematical expression or describe graphically. Hint: You should clearly consider the continuity property of a c.d.f ) --

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