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Chapter 5: Q 5.155. (page 242)
Give two examples of Bernoulli trials other than those presented in the text.
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Playing Cards. An ordinary deck of playing cards has 52 cards. There are four suits-spades, hearts, diamonds, and clubs- with 13 cards in each suit. Spades and clubs are black; hearts and diamonds are red. If one of these cards is selected at random, what is the probability that it is
(a). a spade? (b). red? (c). not a club?
According to the Daily Racing Farm, the probability is about \(0.67\) that the favorite in a horse race will finish in the money (first, second or third place). In the next five races, what is the probability that the favorite finishes in the money.
a. exactly twice?
b. exactly four times
c. at least four times?
d. between two and four times, inclusive?
e. Determine the probability distribution of the random variable \(X\), the number of times the favorite finishes in the money in the next five races.
f. Identify the probability distribution of \(X\) as right skewed, symmetric or left skewed without consulting its probability distribution or drawing its probability histogram.
g. Draw a probability histogram for \(X\).
h. Use your answer from part (c) and definitions \(5.9\) and \(5.10\) on pages \(227\) and \(229\) respectively to obtain the mean and standard deviation of the random variable \(X\).
i. Use formula \(5.5\) on page \(239\) to obtain the mean and standard deviation of the random variable \(X\).
j. Interpret your answer for the mean in words.
Expected Utility. One method for deciding among various investment involves the concept of expected utility. Economists describe the importance of various levels of wealth by using utility functions. For instance, in most case, a single dollar is more important (has greater utility ) for someone with little wealth than for someone with greater wealth Consider two investments, say investment A and B. Measured in thousand of dollars, suppose that investment A yields 0, 1, and 4 with probability 0.1 and16 with probability 0.5, 0.3 and 0.2 respectively. Let Y denote the yield of an investment. For the two investment, determine and compare.
Part (a) The mean of Y, the expected yield.
Part (b) The mean of ,the expected utility, using the utility function role="math" localid="1651845051902" Interpret the utility function
Part (c) The mean of ,the expected utility, using the utility function . Interpret the utility function v
The probability is 0.667 that the favorite in a horse race will finish in the money (first, second, or third place). In 500 horse races, roughly how many times will the favorite finish in the money?
Constract a venn diagram representing the event.
Part (a) (A (not B)).
Part (b) ((A or B) & (not(A & B)))
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