"Multinomial Experiments In Exercises 39 and 40, use the information below.
A multinomial experiment satisfies these conditions.
The experiment has a fixed number of trials n, where each trial is independent of the other trials.
Each trial has k possible mutually exclusive outcomes:
Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is
The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.
The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is





Genetics Another proposed theory in genetics gives the corresponding probabilities for the four types of plants described in Exercise 39 as , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless."

Finding and Interpreting Mean, Variance, and Standard Deviation In Exercises 31–36, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.

Late for Work Thirty-one percent of U.S. employees who are late for work blame oversleeping. You randomly select 12 U.S. employees who are late for work and ask them whether they blame oversleeping. The random variable represents the number who are late for work and blame oversleeping. (Source: CareerBuilder)

Finding and Interpreting Mean, Variance, and Standard Deviation In Exercises 31–36, find the mean, variance, and standard deviation of the binomial distribution for the given random variable. Interpret the results and determine any unusual values.

Life on Other Planets Seventy-nine percent of U.S. adults believe that life on other planets is plausible. You randomly select eight U.S. adults and ask them whether they believe that life on other planets is plausible. The random variable represents the number who believe that life on other planets is plausible. (Source: Ipsos)

Multinomial Experiments In Exercises 39 and 40, use the information below.

A multinomial experiment satisfies these conditions.

The experiment has a fixed number of trials n, where each trial is independent of the other trials.

Each trial has k possible mutually exclusive outcomes:

Each outcome has a fixed probability. So, . The sum of the probabilities for all outcomes is

The number of times occurs is , the number of times occurs is , the number of times occurs is , and so on.

The discrete random variable x counts the number of times that each outcome occurs in n independent trials where . The probability that x will occur is

Genetics According to a theory in genetics, when tall and colorful plants are crossed with short and colorless plants, four types of plants will result: tall and colorful, tall and colorless, short and colorful, and short and colorless, with corresponding probabilities of , and . Ten plants are selected. Find the probability that 5 will be tall and colorful, 2 will be tall and colorless, 2 will be short and colorful, and 1 will be short and colorless.

Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Immigration The mean number of people who immigrated to the United States per hour was about 5.5 in April 2021. Find the probability that the number of people who immigrate to the U.S. in a given hour in April 2021 was (a) zero, (b) exactly five, and (c) exactly eight. (Source: U.S. Census Bureau)

Discrete Variables and Continuous Variables In Exercises 13–18, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the fitted hat sizes of members of a softball team.

Independent and Dependent Random Variables Two random variables x and y are independent when the value of x does not affect the value of y. When the variables are not independent, they are dependent. A new random variable can be formed by finding the sum or difference of random variables. If a random variable x has mean and a random variable y has mean , then the means of the sum and difference of the variables are given by . If random variables are independent, then the variance and standard deviation of the sum or difference of the random variables can be found. So, if a random variable x has variance and a random variable y has variance , then the variances of the sum and difference of the variables are given by In Exercises 43 and 44, the distribution of SAT mathematics scores for college-bound male seniors in 2020 has a mean of 531 and a standard deviation of 121. The distribution of SAT mathematics scores for college-bound female seniors in 2020 has a mean of 516 and a standard deviation of 112. One male and one female are randomly selected. Assume their scores are independent. (Adapted from College Board)

Find the mean and standard deviation of the sum of their scores.