5. Binomial Distribution & Discrete Random Variables

Mean, Variance, and Standard Deviation In Exercises 11–14, find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.

n = 50, p = 0.4

Lightning Deaths Based on the results given in Cumulative Review Exercise 6, assume that for a randomly selected lightning death, there is a 0.8 probability that the victim is a male.

a. Find the probability that three random people killed by lightning strikes are all males.

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).

Hybrids Assume that offspring peas are randomly selected in groups of 16.

b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Acrophobia USA Today reported results from a survey in which subjects were asked if they are afraid of heights in tall buildings. The results are summarized in the accompanying table. Does this table describe a probability distribution? Why or why not?

In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.

In a Pew Research Center survey, 3930 subjects were asked if they have ever fired a gun, and the responses consist of “yes” or “no.”

In Exercises 1–3, 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.

One out of every 42 tax returns for incomes over \$1 million requires an audit. An auditor is examining tax returns for over \$1 million. Find the probability that (c) none of the first five returns the tax auditor examines require an audit.

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

b. Find the probability of exactly 152 yellow peas.

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (c) more than three.

Putting It Together: Naughty or Nice? Yale University graduate student J. Kiley Hamlin conducted an experiment in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed to choose which toy he or she wished to play with. In 14 of the 16 cases, the baby chose the helper toy.

Why is it important to randomly expose the baby to the helper or hinderer toy first?

c. In testing 12 six-month-old babies, all 12 preferred the helper toy. The P-value was reported as 0.0002. Interpret this result.

State the criteria for a binomial probability experiment.

"Consider a binomial probability distribution with parameters n = 5 and p = 0.2.

a. Construct a binomial probability distribution with these parameters."

Salary Negotiations In a Jobvite survey, 2287 adult workers were randomly selected and asked about salary negotiations.

a. 29% of the respondents reported that they negotiated salary at their latest job. What is the number of respondents who reported that they negotiated salary?

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.

Oil Tankers In the month of June 2021, 240 oil tankers stop at a port city. No oil tanker visits more than once. Find the probability that the number of oil tankers that stop on any given day in June is (b) at most three

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.

A biologist is monitoring a large bird sanctuary where a particular bird species is known to have a 70% success rate for each nesting attempt (at least one chick fledges from the nest). This season, she observes 500 independent nesting attempts across the sanctuary.

(C) What is the probability that at least 330 nesting attempts are successful?