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

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{rrr} 0.42 & 0.15 & 0.43 \\0 & 0.73 & 0.27 \\0 & 0 & 1\end{array}\right]$

Evaluate the expression. - ${ }_{6} \mathrm{P}_{5}$

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{rrr} .75 & .25 & 0 \\0 & .9 & .1 \\)2 & .3 & .5\end{array}\right]$

Construct the transition matrix for the transition diagram. -

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: The manufacturer believes that the probability that the new product is better is 0.1 . What is the best strategy?

Prepare a probability distribution for the experiment. Let $x$ represent the random variable, and let $P$ represent theprobability. -Four cards are drawn from a deck. The number of red tens is counted.

Decide whether or not the transition matrix is regular. - $\left[\begin{array}{ll}0.9 & 0.1 \\ 0.4 & 0.6\end{array}\right]$

Solve the problem. -How many distinguishable permutations of letters are possible using the letters in the word COLORADO?

Decide whether or not the transition matrix is regular. - $\left[\begin{array}{cc}1 & 0 \\ 0.2 & 0.8\end{array}\right]$

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{rrr} .80 & .10 & .10 \\)15 & .80 & .05 \\)20 & .70 & .10\end{array}\right]$

Construct the transition matrix for the transition diagram. -

Evaluate the expression. - $5^{\mathrm{P}} 4$

Evaluate the expression. - $10^{P_{2}}$

Find the equilibrium vector for the transition matrix. - $\left[\begin{array}{lll}.1 & .1 & .8 \\ .3 & .3 & .4 \\ .4 & .4 & .2\end{array}\right]$

Construct the transition matrix for the transition diagram. -

For the transition matrix, find the probability that state 2 changes to state 4 after 5 repetitions of the experiment. - $\left[\begin{array}{lllll}0.1 & 0.1 & 0.1 & 0.2 & 0.5 \\ 0.2 & 0.1 & 0.1 & 0.1 & 0.5 \\ 0.1 & 0.2 & 0.3 & 0.2 & 0.2 \\ 0.2 & 0.2 & 0.1 & 0.4 & 0.1 \\ 0.1 & 0.1 & 0.3 & 0.2 & 0.3\end{array}\right]$

Solve the problem. -How many distinguishable permutations of letters are possible using the letters in the word CRITICS?

Solve the problem. -Amy has 3 blue, 2 red, and 5 green books to arrange on a shelf. In how many distinguishable ways can the books be arranged if books of the same color are identical?

Solve the problem. -Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red aces and two black jacks?

Find the probability of the event. -On a hospital floor, 16 patients have a disease with a mortality rate of .1. Two of them die.