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Chapter 5: Q1 (page 195)

The accompanying table lists probabilities for the corresponding numbers of girls in four births. What is the random variable, what are its possible values, and are its values numerical?

Number of Girls x

P(x)

0

0.063

1

0.250

2

0.375

3

0.250

4

0.063

Short Answer

Expert verified

The random variable is x.

The possible values of variable x are 0,1,2,3, and 4.

The variable x is numerical.

Step by step solution

01

Given information

The probabilities for the number of girls in four births are provided.

02

Identify the random variable

A random variable is a variable thatcan take values in the form of numbers.

In the given scenario, the variable that representsthe total number of girls in the four birthsis a random variable, represented by variable x.

Thus, the random variable in the given scenario is x.

03

Provide the possible values of x

The possible values of x are the counts ofthe feasible number of girls.

From the provided table, the possible values that a random variable x (Number of girls) can take are0,1,2,3, and 4.

04

State if the values of x are numerical

The number of girls (x) can be counted and expressed numerically.

Therefore, the values of x are numerical.

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Most popular questions from this chapter

In a study of brand recognition of Sony, groups of four consumers are interviewed. If xis the number of people in the group who recognize the Sony brand name, then xcan be 0, 1, 2, 3, or 4, and the corresponding probabilities are 0.0016, 0.0250, 0.1432, 0.3892, and 0.4096. Does the given information describe a probability distribution? Why or why not?

In Exercises 21–25, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).

Find the mean and standard deviation for the numbers of sleepwalkers in groups of five.

x

P(x)

0

0.172

1

0.363

2

0.306

3

0.129

4

0.027

5

0.002

In Exercises 15–20, refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children.

Using Probabilities for Significant Events

a. Find the probability of getting exactly 1 girl in 8 births.

b. Find the probability of getting 1 or fewer girls in 8 births.

c. Which probability is relevant for determining whether 1 is a significantly low number ofgirls in 8 births: the result from part (a) or part (b)?

d. Is 1 a significantly low number of girls in 8 births? Why or why not?

Number of girls x

P(x)

0

0.004

1

0.031

2

0.109

3

0.219

4

0.273

5

0.219

6

0.109

7

0.031

8

0.004

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.)

In Exercises 15–20, refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children.

Using Probabilities for Significant Events

a. Find the probability of getting exactly 7 girls in 8 births.

b. Find the probability of getting 7 or more girls in 8 births.

c. Which probability is relevant for determining whether 7 is a significantly high number ofgirls in 10 births: the result from part (a) or part (b)?

d. Is 7 a significantly high number of girls in 8 births? Why or why not?

Number of girls x

P(x)

0

0.004

1

0.031

2

0.109

3

0.219

4

0.273

5

0.219

6

0.109

7

0.031

8

0.004
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