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Chapter 3: Q.3.28 (page 109)
Prove or give a counterexample. If and are independent, then they are conditionally independent given .
Rolling two independent six sided dice.
- the number on the first die is , - the number on the second die is , - the sum of the numbers is on both dice is .
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The following method was proposed to estimate the number of people over the age of 50 who reside in a town of known population 100,000: 鈥淎s you walk along the streets, keep a running count of the percentage of people you encounter who are over 50. Do this for a few days; then multiply the percentage you obtain by 100,000 to obtain the estimate.鈥� Comment on this method. Hint: Let p denote the proportion of people in the town who are over 50. Furthermore, let 伪1 denote the proportion of time that a person under the age of 50 spends in the streets, and let 伪2 be the corresponding value for those over 50. What quantity does the method suggest estimate? When is the estimate approximately equal to p?
An urn initially contains 5 white and 7 black balls. Each time a ball is selected, its color is noted and it is replaced in the urn along with 2 other balls of the same color. Compute the probability that (a) the first 2 balls selected are black and the next 2 are white; (b) of the first 4 balls selected, exactly 2 are black.
Suppose that 5 percent of men and 0.25 percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females
Die A has 4 red and 2 white faces, whereas die B has
2 red and 4 white faces. A fair coin is flipped once. If it
lands on heads, the game continues with die A; if it lands on tails, then die B is to be used.
(a) Show that the probability of red at any throw is 12
(b) If the first two throws result in red, what is the probability of red at the third throw?
(c) If red turns up at the first two throws, what is the probability
that it is die A that is being used?
Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \(250,000 per year, while a total of 3.3 percent of all California households earn more than \)250,000 per year
(a) What proportion of all non-California households earn more than \(250,000 per year?
(b) Given that a randomly chosen U.S. household earns more than \)250,000 per year, what is the probability it is a California household
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