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The survival rate during a risky operation for patients with no other hope of survival is \(80 \% .\) What is the probability that exactly four of the next five patients survive this operation?

Of all the trees planted by a landscaping firm, \(90 \%\) survive. What is the probability that 8 or more of the 10 trees they just planted will survive? (Find the answer by using a table.)

In the biathlon event of the Olympic Games, a participant skis cross-country and on four intermittent occasions stops at a rifle range and shoots a set of five shots. If the center of the target is hit, no penalty points are assessed. If a particular man has a history of hitting the center of the target with \(90 \%\) of his shots, what is the probability of the following? a. He will hit the center of the target with all five of his next set of five shots. b. He will hit the center of the target with at least four of his next set of five shots. (Assume independence.)

A January 2005 survey of bikers, commissioned by the Progressive Group of Insurance Companies, showed that \(40 \%\) of bikers have body art, such as tattoos and piercings. A group of 10 bikers are in the process of buying motorcycle insurance. a. What is the probability that none of the 10 has any body art? b. What is the probability that exactly 3 have some body art? c. What is the probability that at least 4 have some body art? d. What is the probability that no more than 2 have some body art?

Find the mean and standard deviation for the binomial random variable \(x\) with \(n=30\) and \(p=0.6,\) using formulas (5.7) and (5.8).

Find the mean and standard deviation of \(x\) for each of the following binomial random variables: a. The number of tails seen in 50 tosses of a quarter b. The number of left-handed students in a classroom of 40 students (Assume that \(11 \%\) of the population is left-handed.) c. The number of cars found to have unsafe tires the 400 cars stopped at a roadblock for inspection (Assume that \(6 \%\) of all cars have one or more unsafe tires.) d. The number of melon seeds that germinate when a package of 50 seeds is planted (The package states that the probability of germination is 0.88.)

A binomial random variable \(x\) is based on 15 trials with the probability of success equal to 0.2. Find the probability that this variable will take on a value more than 2 standard deviations from the mean.

What are the two basic properties of every probability distribution?

a. Explain the difference and the relationship between a probability distribution and a probability function. b. Explain the difference and the relationship between a probability distribution and a frequency distribution, and explain how they relate to a population and a sample.

"How many TVs are there in your household?" was one of the questions on a questionnaire sent to 5000 people in Japan. The collected data resulted in the following distribution: $$\begin{array}{l|cccccc}\hline \text { Number of TVs/Household } & 0 & 1 & 2 & 3 & 4 & 5 \text { or more } \\\\\text { Percentage } & 1.9 & 31.4 & 23.0 & 24.4 & 13.0 & 6.3 \\\\\hline\end{array}$$ a. What percentage of the households have at least one television? b. What percentage of the households have at most three televisions? c. What percentage of the households have three or more televisions? d. Is this a binomial probability experiment? Justify your answer. e. Let \(x\) be the number of televisions per household. Is this a probability distribution? Explain. f. Assign \(x=5\) for "5 or more" and find the mean and standard deviation of \(x.\)