Chapter 5: Probability and Random Variables

Evaluate the following binomial coefficients.

$\mathrm{a}.\left(\begin{array}{c}5\\ 3\end{array}\right)\mathrm{b}.\left(\begin{array}{c}10\\ 0\end{array}\right)\mathrm{c}.\left(\begin{array}{c}10\\ 10\end{array}\right)\mathrm{d}.\left(\begin{array}{c}9\\ 5\end{array}\right)$

For each of the following probability histograms of binomial distributions, specify whether the success probability is less than, equal to, or greater than 0.5. Explain your answers.

For each of the following probability histograms of binomial distributions, specify whether the success probability is less than, equal to, or greater than 0.5. Explain your answers.

Pinworm Infestation. Pinworm infestation, which is commonly found in children, can be treated with the drug pyrantel pamoute. According to the Menk Mamual, the treatment is effective in 90 % of cases. Suppose that three children with pinworm infestation are given pyrantel pamoate.

a. Considering a success in a given case to be "a cure," formulate the process of observing which children are cured and which children are not cured as a sequence of three Bernoulli trials.

b. Constract a table similar to Table \(5.19\) on page 234 for the three cases. Display the probabilities to three decimal places.

c. Draw a tree diagram for this problem similar to the one shown in Fig. \(5.24\) on page 235 .

d. List the outcomes in which exactly two of the three children are cured.

e. Find the probability of each outcome in part (d). Why are those probabilities all the same?

f. Use parts (d) and (e) to determine the probability that exactly two of the three children will be cured.

g. Without using the binomial probability formula, obtain the probability distribution of the random variable \(X\), the number of children out of three who are cured.

Psychiatric Disorders. The National Institute of Mental Health reports that there is a 20 % chance of an adult American suffering from a psychiatric disorder. Four randomly selected adult Americans are examined for psychiatric disorders.

a. If you let a success correspond to an adult American having a psychiatric disorder, what is the success probability. $p$? (Note: The use of the word success in Bernoulli trials need not reflect its usually positive connotation.)

b. Construct a table similar to Table \(5.19\) on page 234 for the four people examined. Display the probabilities to four decimal places.

c. Draw a tree diagram for this problem similar to the one shown in Fig. \(5.24\) on page 235 .

d. List the outcomes in which exactly three of the four people examined have a psychiatric disorder.

e. Find the probability of each outcome in part (d). Why are those probabilities all the same?

f. Use parts (d) and (e) to determine the probability that exactly three of the four people examined have a psychiatric disorder.

g. Without using the binomial probability formula, obtain the probability distribution of the random variable $Y$, the number of adults out of four who have a psychiatric disorder.

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. Let X denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) Table VII in Appendix A. Compare your answer here to that in part (a).

$n=4,p=0.3,P(X=2)$

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. LetX denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) TableVII in AppendixA. Compare your answer here to that in part (a).

$n=5,p=0.6,P(X=3)$

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. LetX denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) TableVII in AppendixA. Compare your answer here to that in part (a).

$n=6,p=0.5,P(X=4)$

In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.

Russian Presidential Election. According to the Central Election Commission of the Russian Federation, a frequency distribution for the March 4. 2012 Russian presidential election is as follows.

Find the probability that a randomly selected voter voted for

a. Putin.

b. either Zhirinovsky or Mironov.

c. someone other than Putin.

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. LetX denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) TableVII in AppendixA. Compare your answer here to that in part (a).

$n=3,p=0.4,P(X=1)$