Exam 9: Counting, Probability Distributions, and Further Topics in Probability

Decide whether or not the matrix is a probability vector. - $\left[\begin{array}{llll}0.11 & 0.28 & 0.39 & 0.22\end{array}\right]$

Use the payoff matrix to determine the best strategy. -A computer manufacturer must decide whether or not to market a new product. The new product may or may not be better than the old product. If they market the new product and it is better than the old product, their sales should increase. If they market the new product and it is not better, they will lose money to competitors. If they do not market the new product, they will lose to competitors if it is actually better and will lose just the research costs if it is not better. The manufacturer estimates that the payoff matrix is as follows: What should the manufacturer do if he is an optimist?

Decide whether or not the matrix is a probability vector. - $\left[\begin{array}{ll}0.8 & 0.8\end{array}\right]$

Find the expected value of the optimal strategy. -A farmer must decide on which of two pieces of land to grow his crops. The first piece of land has better soil and will yield a larger harvest. However this piece of land is low lying and is coastal and his crops will be destroyed in the event of a storm and flooding. The second piece of land has poorer soil and will yield a smaller harvest but is at a higher elevation and would not be affected by flooding. He estimates that the payoff matrix is as follows: Based on meteorological records from previous years, the farmer estimates the probability of flooding next year to be 0.15 . What is the farmer's expected profit if he picks the best strategy?

Decide whether or not the matrix is a probability vector. - $\left[\begin{array}{lll}\frac{1}{2} & \frac{3}{8} & \frac{1}{6}\end{array}\right]$

Sketch a transition diagram for the given transition matrix. - $\left[\begin{array}{rr} 0.5 & 0.5 \\1 & 0\end{array}\right]$

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability. -A field goal kicker has a kicking average of .75 and he tries 3 field goals in a game. The number of field goals is counted.

Find the expected value for the random variable. -For a certain animal species, the probability that a female will have a certain number of offspring in a given year is given in the table below. Find the expected number of offspring per year.

Find the expected value for the random variable. -

Evaluate the expression. - ${ }_{23} \mathrm{C}_{14}$

A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting the given result. -Exactly five twos

Solve the problem. -Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if the men are to receive their awards first, and then the women?

Find the expected value for the random variable. -

Solve the problem. -Find the number of ways of arranging 5 distinct items in a circle.

Solve the problem. -Suppose you pay $\$ 1.00$ to roll a fair die with the understanding that you will get back $\$ 3$ for rolling a 1 or a 5 , nothing otherwise. What are your expected winnings?

Find the requested long-range probabilities based on the transition matrix or data given. -Weather is classified as sunny or cloudy in a certain place. The probability that it will be sunny on a given day depends on whether it was sunny the previous day. The transition matrix is given below. Find the long-range prediction for the proportion of sunny and cloudy days.

Decide whether or not the matrix is a probability vector. - $\left[\begin{array}{ll}\frac{3}{10} & \frac{7}{10}\end{array}\right]$

Give the probability distribution and sketch the histogram. -The telephone company kept track of the calls for the correct time during a 24 -hour period for two weeks. The results are shown in the table.

Provide an appropriate response. -Suppose that a class of 30 students is assigned to write an essay. 1) Suppose 4 essays are randomly chosen to appear on the class bulletin board. How many different groups of 4 are possible? 2) Suppose 4 essays are randomly chosen for awards of $10, $7, $5, and $3. How many different groups of 4 are possible? Explain the significant differences between problems 1 and 2.

Solve the problem. -Find the number of ways of arranging 8 distinct items in a circle.