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

Find the expected value for the random variable. -A business bureau gets complaints as shown in the following table. Find the expected number of complaints per day.

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 four twos

Decide whether or not the matrix is a transition matrix. - $\left[\begin{array}{ll}\frac{4}{5} & \frac{1}{5} \\\frac{3}{9} & \frac{6}{9}\end{array}\right]$

Provide an appropriate response. -Consider the following formulas: ${ }_{n} \mathrm{P}_{\mathrm{r}}=\frac{\mathrm{n} \text { ! }}{(n-r) !}$ and ${ }_{n} \mathrm{C}_{\mathrm{r}}=\frac{n !}{(n-r) ! r !}$ . Given the same values for $\mathrm{n}$ and $\mathrm{r}$ in each formula, which is the smaller value, $\mathrm{P}$ or $\mathrm{C}$ ? How does this relate to the concept of counting the number of outcomes based on whether or not order is a criterion?

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{ll} \frac{1}{4} & \frac{3}{4} \\\frac{1}{2} & \frac{1}{2}\end{array}\right]$

Use the multiplication principle to solve the problem. -License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Decide whether or not the matrix is a probability vector. - $\left[\begin{array}{llll}0 & 0.5 & 0.4 & 0.1\end{array}\right]$

Find the expected value of the random variable in the experiment. -A bag contains six marbles, of which four are red and two are blue. Suppose two marbles are chosen at random and $\mathrm{X}$ represents the number of red marbles in the sample.

Find the expected value for the random variable x having this probability function. - =5 =6 =7 =8

A bag contains 6 cherry, 3 orange, and 2 lemon candies. You reach in and take 3 pieces of candy at random. Find the probability. -1 cherry, 2 lemon

Find the requested long-range probabilities based on the transition matrix or data given. -Tulips with one red and one white gene produce pink tulips. Find the long-range prediction for the fraction of red, pink and white tulips.

Solve the problem. -Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex?

Solve the problem. -Three student representatives, a president, a secretary, and a treasurer, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the president must be a woman and the secretary and treasurer must be men?

Prepare a payoff matrix -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 but is at a higher elevation and would not be affected by flooding. If he uses the low-lying piece of land and there is no flooding he will make a profit of $\$ 13,000$ . If he uses the low-lying piece of land and there is flooding, he will lose $\$ 5000$ . If he uses the higher piece of land and there is no flooding, he will make a profit of $\$ 10,000$ . If he uses the higher piece of land and there is flooding, he will make a profit of $\$ 14,000$ . (If there is flooding there will be a smaller supply of the crop and his profit will be higher). Prepare a payoff matrix.

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{ll} \frac{2}{3} & \frac{1}{3} \\\frac{1}{4} & \frac{3}{4}\end{array}\right]$

Provide an appropriate response. -In the context of Markov chains, explain why the matrix below could not be a transition matrix. $\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{3} \\ \frac{1}{4} & \frac{3}{4}\end{array}\right]$

Find the requested probability. -What is the probability that 16 rolls of a fair die will show 5 fours?

Solve the problem. -10 people are bowling: 2 use a black ball and 8 use a blue ball. How many different black and blue bowling ball sequences can occur on the rack?

Provide an appropriate response. -In the context of Markov chains, explain why the matrix below could not be a transition matrix. $\left[\begin{array}{ccc}\frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1\end{array}\right]$

Solve the problem. -Suppose 6 people sit at a circular table. Find the probability that 2 particular people are sitting next to each other.