Probability

   Suppose you go out for pizza with two friends. You have agreed to the following rule to decide who will pay the bill. Each person will toss a coin. The person who gets a result that is diGGerent from the other two will pay the bill. If all three tosses yield the same result, the bill will be shared by all. B List the sample space. (Hint: What are the possible combinations of the coin toss?) C Find the probability that only you will have to pay. D Find the probability that all three will share the expense.  Suppose "
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P(A)=.50 and P(B)=.22. Answer the following questions. (a) Determine P(A S B) if A and B are independent. (b) Determine P(A S B) if A and B are mutually exclusive. . Let X be a random variable that represents the number of students who are absent on a given day from a class of 25. The following table lists the probability distribution of X. x 012345 P(x) .08 .18 .32 .22 .14 .06 (a) Determine the following probabilities. P(X = 4), P(X > 4), P(2 < X  4), and P(X 1). (b) Determine the expected number of absent students on a given day. (c) Compute the variance of X by definition. 
According to the Mendelian theory of inherited characteristics, a cross fertilization of related spices of red and yellow flowered plants produces a generation whose oGGTpring contain 25% red-flowered plants. Suppose that a horticulturist wishes to cross 5 pairs of the cross-fertilized species. B Justify that each cross fertilization of the red- and yellow-flowered plants is a Bernoulli trial. Let X be the number of red-flowered plants in 5 pairs of the cross-fertilized species. C Is X a binomial random variable? If Yes, identify n and p. D Determine the probability that there will be no red-flowered plant. E Determine the probability that there will be at least one red-flowered plant. F Determine E(X), V ar(X) and SD(X). (Hint:not necessary to compute through the definition of mean and variance.) (d) Determine the standard deviation of X. 
Z is a standard normal random variable. Determine the following probabilities. B P (0 < Z < 1.96), P (Z < 1.96), and P (Z < 1.96). C P (1.5 < Z < 2), and P (1.5 < Z < 2). D P (1 < Z < 1), P (2 < Z < 2), and P (3 < Z < 3). Compare these probabilities to those in the empirical rule.