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Chapter 5: Q10 (page 195)

In Exercises 7–14, determine whether a probability

distribution is given. If a probability distribution is given, find its mean and standarddeviation. If a probability distribution is not given, identify the requirements that are notsatisfied.

In a Microsoft Instant Messaging survey, respondents were asked to choose the most fun way to flirt, and the accompanying table is based on the results.

x

P(x)

E-mail

0.06

In person

0.55

Instant message

0.24

Text message

0.15

Short Answer

Expert verified

The requirement that x is a numerical random variable is not satisfied.

Step by step solution

01

Given information

The probability distribution for the most fun way to flirt is provided.

02

Identify the requirements for a probability distribution

Letthe variable x represent the type of messaging.

The requirements are as follows.

1)The variablex is a numerical random variable,which is not satisfied as the variable cannot be expressed in the number form.

2)The sum of the probabilities is computed as

∑Px=0.06+0.55+0.24+0.15=1

Therefore, the sum of the probabilities is 1.

3) Each value of P(x) is between 0 and 1.

Thus, the requirement that x is a numerical random variable is not satisfied.

Therefore, the mean and standard deviation cannot be computed.

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Most popular questions from this chapter

In Exercises 5 and 6, refer to the given values, then identify which of the following is most appropriate:discrete randomvariable, continuous random variable, ornot a random variable.

a. Exact weights of the next 100 babies born in the United States

b. Responses to the survey question “Which political party do you prefer?”

c. Numbers of spins of roulette wheels required to get the number 7

d. Exact foot lengths of humans

e. Shoe sizes (such as 8 or 8½) of humans

Hypergeometric Distribution If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type (such as lottery numbers you selected), while the remaining B objects are of the other type (such as lottery numbers you didn’t select), and if n objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting x objects of type A and n - x objects of type B is

\(P\left( x \right) = \frac{{A!}}{{\left( {A - x} \right)!x!}} \times \frac{{B!}}{{\left( {B - n + x} \right)!\left( {n - x} \right)!}} \div \frac{{\left( {A + B} \right)!}}{{\left( {A + B - n} \right)!n!}}\)

In New Jersey’s Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probabilities of getting exactly two winning numbers with one ticket. (Hint: Use A = 6, B = 43, n = 6, and x = 2.)

Expected Value in Roulette When playing roulette at the Venetian casino in Las Vegas, a gambler is trying to decide whether to bet \(5 on the number 27 or to bet \)5 that the outcome is any one of these five possibilities: 0, 00, 1, 2, 3. From Example 6, we know that the expected value of the \(5 bet for a single number is -26¢. For the \)5 bet that the outcome is 0, 00, 1, 2, or 3, there is a probability of 5/38 of making a net profit of \(30 and a 33/38 probability of losing \)5.

a. Find the expected value for the \(5 bet that the outcome is 0, 00, 1, 2, or 3.

b. Which bet is better: a \)5 bet on the number 27 or a $5 bet that the outcome is any one of the numbers 0, 00, 1, 2, or 3? Why?

a.The probability of 7 hurricanes in a year is equal to 0.140.

b. Thus, the expected number of years to have 7 hurricanes in a 55-year period is equal to 7.7 years.

c. The expected number of years that have 7 hurricanes is approximately equal to the actual number of years that have 7 hurricanes in a 55-year period.Since the expected and the actual values are approximately equal, the Poisson distribution works well here.

In Exercises 15–20, assume that random guesses are made for eight multiple choice questions on an SAT test, so that there are n = eight trials, each with probability of success (correct) given by p = 0.20. Find the indicated probability for the number of correct answers.

Find the probability that the number x of correct answers is no more than 2.
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