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Chapter 5: Q 5.86. (page 216)
Suppose that A and B are events such that ,and
Part (a). Are event A and B mutually exclusive ? Explain your answer.
Part (b) Find
Part (a) and are not mutually exclusive because and are not zero.
Part (b)
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Archery. An archer shoots an arrow into a square target 6 feet on a side whose center we call the origin. The outcome of this random experiment is the point in the target hit by the arrow. The archer scores 10 points if she hits the bull's eye-a disk of radius 1 foot centered at the origin; she scores 5 points if she hits the ring with inner radius 1 foot and outer radius 2 feet centered at the origin; and she scores 0 points otherwise. Assume that the archer will actually hit the target and is equally likely to hit any portion of the target. For one arrow shot, let S be the score.
(a) Obtain and interpret the probability distribution of the random variable S. (Hint: The area of a square is the square of its side length; the area of a disk is the square of its radius times.)
(b) Use the special addition rule and the probability distribution obtained in part (a) to determine and interpret the probability of each of the following events:
Dice. Refer to exercise 5.53.
a Are events A and B mutually exclusive?
b Are events B and C mutually exclusive?
c Are events A, C and D mutually exclusive?
d Are there three mutually exclusive events among A, B, C and D? four?
A variable y of a finite population has the following frequency distribution:
| y | 0 | 1 | 4 | 6 |
| f | 18 | 14 | 8 | 10 |
Suppose a member is selected at random from the population and let Y denote the value of the variable y for the member obtained.
a. Determine the probability distribution of the random variable Y.
b. Use random-variable notation to describe the events that Y takes on the value 3, a value less than 3, and a value of at least 3.
c. Find P(Y = 3), P(Y < 3), and P(Y 3). Interpret your results.
d. Construct a probability histogram for the random variable Y.
Normality Requirement What is different about the normality requirement for a confidenceinterval estimate of \(\sigma \)and the normality requirement for a confidence interval estimateof \(\mu \)?
Video Games. A pathological video game user (PVGU) is a video game user that averages 31 or more hours a week of gameplay.
According to the article "Pathological Video Game Use among Youths: A Two-Year Longitudinal Study" (Pediatrics, Vol. 127. No. 2, pp. 319-329) by D. Gentile et al., in 2011, about 9% of children in grades 3-8 were PVGUs. Suppose that, today, seven youths in grades 3-8 are randomly selected.
(a) Assuming that the percentage of PVGUS in grades 3-8 is the same today as it was in 2011, determine the probability distribution for the number, X, who are PVGUs.
(b) Determine and interpret the mean of X.
(c) If, in fact, exactly three of the seven youths selected are PVGUs, would you be inclined to conclude that the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning. Hint: First consider the probability .
(d) If, in fact, exactly two of the seven youths selected are PVGUs, would you be inclined to conclude the percentage of PVGUs in grades 3-8 has increased from the 2011 percentage? Explain your reasoning.
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