91Ó°ÊÓ

The following table gives the probability distribution of a discrete random variable \(x\). $$ \begin{array}{l|ccccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P(x) & .11 & .19 & .28 & .15 & .12 & .09 & .06 \\ \hline \end{array} $$ Find the following probabilities. a. \(P(3)\) b. \(P(x \leq 2)\) c. \(P(x \geq 4)\) d. \(P(1 \leq x \leq 4)\) e. Probability that \(x\) assumes a value less than 4 f. Probability that \(x\) assumes a value greater than 2 g. Probability that \(x\) assumes a value in the interval 2 to 5

A review of emergency room records at rural Millard Fellmore Memorial Hospital was performed to determine the probability distribution of the number of patients entering the emergency room during a 1 -hour period. The following table lists this probability distribution. $$ \begin{array}{l|ccccccc} \hline \begin{array}{l} \text { Patients } \\ \text { per hour } \end{array} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .2725 & .3543 & .2303 & .0998 & .0324 & .0084 & .0023 \\ \hline \end{array} $$ a. Make a histogram for this probability distribution. b. Determine the probability that the number of patients entering the emergency room during a randomly selected 1 -hour period is i. 2 or more ii. exactly 5 iii. fewer than 3 iv. at most 1

One of the most profitable items at Al's Auto Security Shop is the remote starting system. Let \(x\) be the number of such systems installed on a given day at this shop. The following table lists the frequency distribution of \(x\) for the past 80 days. $$ \begin{array}{l|ccccc} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f & 8 & 20 & 24 & 16 & 12 \\ \hline \end{array} $$ a. Construct a probability distribution table for the number of remote starting systems installed on a given day. b. Are the probabilities listed in the table of part a exact or approximate probabilities of various outcomes? Explain. c. Find the following probabilities. i. \(P(3)\) ii. \(P(x \geq 3)\) iii. \(P(2 \leq x \leq 4)\) iv. \(P(x<4)\)

According to a survey, \(30 \%\) of adults are against using animals for research. Assume that this result holds true for the current population of all adults. Let \(x\) be the number of adults who are against using animals for research in a random sample of two adults. Obtain the probability distribution of \(x\). Draw a tree diagram for this problem.

In a group of 12 persons, 3 are left-handed. Suppose that 2 persons are randomly selected from this group. Let \(x\) denote the number of left-handed persons in this sample. Write the probability distribution of \(x\). You may draw a tree diagram and use it to write the probability distribution. (Hint: Note that the selections are made without replacement from a small population. Hence, the probabilities of outcomes do not remain constant for each selection.)

Briefly explain the concept of the mean and standard deviation of a discrete random variable.

The following table gives the probability distribution of the number of camcorders sold on a given day at an electronics store. $$ \begin{array}{l|ccccccc} \hline \text { Camcorders sold } & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \text { Probability } & .05 & .12 & .19 & .30 & .20 & .10 & .04 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution. Give a brief interpretation of the value of the mean.

Let \(x\) be the number of houses sold per month by a real estate agent. The following table lists the probability distribution of \(x\). $$ \begin{array}{l|cccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(x) & .08 & .12 & .32 & .28 & .12 & .08 \\ \hline \end{array} $$ Calculate the mean and standard deviation of this probability distribution and give a brief interpretation of the value of the mean.

Which of the following are binomial experiments? Explain why. a. Rolling a die 10 times and observing the number of spots b. Rolling a die 12 times and observing whether the number obtained is even or odd c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when \(54 \%\) of all voters are known to be in favor of this proposition.

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(5)\) for \(n=8\) and \(p=.70\) b. \(P(3)\) for \(n=4\) and \(p=.40\) c. \(P(2)\) for \(n=6\) and \(p=.30\) Verify your answers by using Table I of Appendix \(\mathrm{B}\).