Exam 8: Probability

You are choosing a region of the country to move to. Let E be the event that the region you choose saw an increase in housing prices of 15% or less, but greater than 10%, and let F be the event that the region you choose is on the west coast. Choose the correct statement.

A dodecahedral die has 12 faces, numbered 1-12. If the die is weighted in such a way that 2 is twice as likely to be facing up as 1, 3 is three times as likely to be facing up as 1,and so on, what is the probability that 7 will land facing up

You are given a transition matrix P and initial distribution vector v. Find the two-step transition matrix and the distribution vector after two steps. $P = \left[ \begin{array} { c c c } 0.1 & 0.9 & 0 \\0 & 1 & 0 \\0 & 0.2 & 0.8\end{array} \right]$ , $v = \left[ \begin{array} { l l l } 0.5 & 0 & 0.5\end{array} \right]$

In a (fictitious) soccer tournament involving the four teams San Diego State, De Paul, Colgate, and Hofstra, find the probability that Hofstra will play De Paul in the finals and win. (Assume that all outcomes are equally likely and that the teams not listed in the first round slots are placed at random.)

The following diagram shows the movement of U.S. households among three income groups - affluent, middle class, and poor - over the 11-year period 1982-1993. ? $a = 26.5 \%$ , $b = 7.8 \%$ , $c = 31.2 \%$ , $d = 8.1 \%$ ? Use the transitions shown in the diagram to construct a transition matrix (assuming zero probabilities for the transitions between affluent and poor). Take 1 = Affluent, 2 = Middle class, 3 = Poor. Assuming that the trend shown was to continue, what percent of households classified as affluent in 1982 were predicted to become poor in 2004 (Give your answer to the nearest 0.1%.) The answer is __________% According to the model, what percentage of all U.S. households will be in each income bracket in the long term (Give your answers to the nearest 0.1 percent.) Percent of affluent is __________% Percent of middle class is __________% Percent of poor is __________%

Student admissions are based on the following table, which shows the profile, by Math SAT I scores, of admitted students at some university for the Fall 2004 semester. Determine the theoretical probability of the event: "An applicant who had a Math SAT of 700 or above was admitted." (Round your answer to the nearest 0.01). - - - - - Total Admitted 10 212 1,132 2,882 5,347 9,583 Not Admitted 675 3,500 8,833 12,230 5,150 30,388 ​ Enter your answer in decimal form, not as a fraction. The probability is __________.

Describe the sample space S of the experiment and list the elements of the given event E. Two indistinguishable dice are rolled; both numbers are composite.

You are deciding on whether to enroll in Psychology 200, Finite Math 400, Greek Poetry 300, or Calculus 100. Describe the event E of avoiding math classes.

Lance the Wizard has been informed that tomorrow there will be a 45% chance of encountering the evil Myrmidons and a 20% chance of meeting up with the dreadful Balrog. Moreover, Hugo the Elf has predicted that there is a 12% chance of encountering both tomorrow. What is the probability that Lance will be lucky tomorrow and encounter neither the Myrmidons nor the Balrog ?

Concerning Internet use In 1999, 59% of all Caucasians in the U.S., 60% of all African-Americans, 58% of all Hispanics, and 54% of residents not classified into one of these groups used the Internet to search for information. At that time, the U.S. population was 69% Caucasian, 12% African-American, and 13% Hispanic. What percentage of U.S. residents who used the Internet for information search were African-American Express your answer to the nearest 1%.

The graph consists of a collection of nodes (the dots in the figure) connected by edges (line segments from one node to another). A move on a graph is a move from one node to another along the single edge. Find the probability of moving from Start to Finish in a sequence of two random moves in the graph. (All directions are equally likely.)

Let W be the event that you will use the website tonight, Q be the event that your math grade will improve, and E be the event that you will use the website every night. Express the following event in symbols. ​ "Either your math grade will not improve, or you will use the website tonight, but not every night." ​ Use the symbol " ~ " for the complement of the event. For example, complement of the event E would be $\sim E$ . ​ ∼

Two of the mathematics professors at Enormous State are Professor A (known for easy grading) and Professor F (known for tough grading). Last semester, roughly half of Professor F's class consisted of former students of Professor A; these students apparently felt encouraged by their (utterly undeserved) high grades. (Professor F's own former students had fled in droves to Professor A's class to try to shore up their grade point averages.) At the end of the semester, as might have been predicted, all of Professor A's former students wound up with a C- or lower. The rest of the students in the class--former students of Professor F who had decided to "stick it out"-fared better, and three quarters of them earned higher than a C-. After discovering what had befallen them, all the students who earned C- or lower got together and decided to send a delegation to the Department Chair to complain that their grade point averages had been ruined by this callous and heartless beast! The contingent was to consist of 10 representatives selected at random from among them. How many of the 10 would you estimate to have been former students of Professor A The answer is __________.

Let W be the event that you will use the website tonight, Q be the event that your math grade will improve, and E be the event that you will use the website every night. Express the following event in symbols. ​ "Either you will use the website every night, or your math grade will improve." ​ Use the symbol " ~ " for the complement of the event. For example, complement of the event E would be $\sim E$ .

In a Paris daily lottery, a sequence of 3 (not necessarily different) digits in the range 0-6 is selected at random. Find the probability that 2 of the 3 digits are the same.

Use the given transition matrix P and the given initial distribution vector v to obtain the two-step transition matrix and the distribution vector after two steps. $P = \left[ \begin{array} { c c } 1 & 0 \\0.4 & 0.6\end{array} \right]$ , $v = \left[ \begin{array} { l l } 0.6 & 0.4\end{array} \right]$

Two indistinguishable dice are rolled. What is the probability of the event $\{ ( 2,3 ) , ( 4,6 ) , ( 5,6 ) \}$ ?

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places. $P ( A \mid B ) = 0.8$ , $P ( B ) = 0.2$ , $P \left( A \mid B ^ { \prime } \right) = 0.3$ . Find $P ( B \mid A )$ .

You are given a transition matrix P. Find the steady-state distribution vector. ​ $P = \left[ \begin{array} { c c c } \frac { 2 } { 3 } & \frac { 1 } { 3 } & 0 \\\frac { 2 } { 3 } & \frac { 1 } { 3 } & 0 \\\frac { 2 } { 3 } & 0 & \frac { 1 } { 3 }\end{array} \right]$

Which of the following represents the transition matrix for the state diagram below. $A = \frac { 3 } { 4 }$ , $B = \frac { 1 } { 4 }$ , $C = \frac { 3 } { 5 }$ , $D = \frac { 2 } { 5 }$