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Chapter 5: Q 5.5. (page 200)
An experiment has 20 possible outcomes, all equally likely. An event can occur in five ways. The probability that the event occurs is .
The probability that the event occurs is.
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In Exercise 5.129 - 5.132, We have provided the probability distributions of the random variables considered in Exercise 5.109 - 5.112 of section 5.4 For each exercise, do the following tasks.
Part (a) Find the mean of the random variable.
Part (b) Obtain the standard deviation of the random variable by using one of the formulas given in definition 5.10.
The Geometric Distribution. In this exercise, we discuss the geometric distribution, the probability distribution for the number of trials until the first success in Bernoulli trials. The geometric probability formula is
where Xdenotes the number of trials until the first success and pthe success probability. Using the geometric probability formula and Definition 5.9 on page 227. we can show that the mean of the random variable Xis 1/p.
To illustrate, consider the Mega Millions lottery, a multi-state jackpot draw game with a jackpot starting at $15 million and growing until someone wins. In order to play, the player selects five white numbers from the numbers 1-75 and one Mega Ball number from the numbers 1-15. Suppose that you buy one Mega Millions ticket per week. Let Xdenote the number of weeks until you win a prize.
(a) Find and interpret the probability formula for the random variable X. (Note: The probability of winning a prize with a single ticket is 0.0680.)
(b) Compute the probability that the number of weeks until you win a prize is exactly 3; at most 3: at least 3.
(c) On average, how long will it be until you win a prize?
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?
Space Shuttles. The National Aeronautics and Space Administration (NASA) compiles data on space-shuttle launches and publishes them on its website. The following table displays a frequency distribution for the number of crew members on each shuttle mission from April 12, 1981 to July 8, 2011.
| Crew Size | 2 | 4 | 5 | 6 | 7 | 8 |
| Frequency | 4 | 3 | 36 | 28 | 63 | 1 |
Let X denote the crew size of a randomly selected shuttle mission between the aforementioned dates.
a. What are the possible values of the random variable X?
b. Use random-variable notation to represent the event that the shuttle mission obtained has a crew size of 7.
c. Find P(X = 4); interpret in terms of percentages.
d. Obtain the probability distribution of X.
e. Construct a probability histogram for X.
Vitamin C and Aspirin A bottle contains a label stating that it contains Spring Valley pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains Bayer pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin C claim or rejection of the Bayer aspirin claim? Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?
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