91Ó°ÊÓ

In exercise, you are performing 5 independent Bernoulli trials with \(p=.1\) and \(q=.9 .\) Calculate the probability of each of the stated outcomes. Check your answer using technology. All failures

If you roll a die 100 times, what is the approximate probability that you will roll between 15 and 20 ones, inclusive? (Round your answer to two decimal places.) HINT [See Example 4.]

If you roll a die 200 times, what is the approximate probability that you will roll fewer than 25 ones, inclusive? (Round your answer to two decimal places.)

Calculate the expected value of the given random variable \(X .\) [Exercises \(23,24,27\), and 28 assume familiarity with counting arguments and probability (Section 7.4).] \(\bigvee X\) is the lower number when two dice are rolled.

Calculate the expected value of the given random variable \(X .\) [Exercises \(23,24,27\), and 28 assume familiarity with counting arguments and probability (Section 7.4).] \(\nabla X\) is the number of green marbles that Suzan has in her hand after she selects four marbles from a bag containing three red marbles and two green ones.

Calculate the expected value of the given random variable \(X .\) [Exercises \(23,24,27\), and 28 assume familiarity with counting arguments and probability (Section 7.4).] \(\nabla 30\) darts are thrown at a dartboard. The probability of hitting a bull's-eye is \(\frac{1}{5}\). Let \(X\) be the number of bull's-eyes hit.

Y o u r ~ p i c k l e ~ c o m p a n y ~ r a t e s ~ i t s ~ p i c k l e s ~ o n ~ a ~ s c a l e ~ o f spiciness from 1 to \(10 .\) Market research shows that customer preferences for spiciness are normally distributed, with a mean of \(7.5\) and a standard deviation of 1 . Assuming that you sell 100,000 jars of pickles, how many jars with a spiciness of 9 or above do you expect to sell?

Pastimes A survey of all the students in your school yields the following probability distribution, where \(X\) is the number of movies that a selected student has seen in the past week: \begin{tabular}{|r|c|c|c|c|c|} \hline Number of Movies & 0 & 1 & 2 & 3 & 4 \\ \hline Probability & \(.5\) & \(.1\) & \(.2\) & \(.1\) & \(.1\) \\ \hline \end{tabular} Compute the expected value \(\mu\) and the standard deviation \(\sigma\) of \(X\). (Round answers to two decimal places.) For what percentage of students is \(X\) within two standard deviations of \(\mu\) ?

In March 2004, the U.S. Agriculture Department announced plans to test approximately 243,000 slaughtered cows per year for mad cow disease (bovine spongiform encephalopathy). \(^{17}\) When announcing the plan, the Agriculture Department stated that "by the laws of probability, that many tests should detect mad cow disease even if it is present in only 5 cows out of the 45 million in the nation." \(^{18}\) Test the Department's claim by computing the probability that, if only 5 out of 45 million cows had mad cow disease, at least 1 cow would test positive in a year (assuming the testing was done randomly).

A roulette wheel has the numbers 1 through 36,0 , and \(00 .\) A bet on two numbers pays 17 to 1 (that is, if one of the two numbers you bet comes up, you get back your \(\$ 1\) plus another \(\$ 17\) ). How much do you expect to win with a \(\$ 1\) bet on two numbers? HINT [See Example 4.]