Exam 8: Probability

Suppose two dice (one green, one red) are rolled. Consider the following events. A: The red die shows 1. B: The numbers add to 6. C: At least one of the numbers is 1. D: The numbers do not add to 11. Express the following in symbols. ​ "The numbers do not add to 6."

The following table shows the results of a survey of 400 authors by a (fictitious) publishing company: New Authors Established Authors Total Successful 60 50 110 Unsuccessful 130 160 290 Total 190 210 400 An author is chosen at random. Compute the estimated probability of the event that the author is established.

Write down the transition matrix associated with the given state transition diagram below. ​ ​ $A = \frac { 1 } { 2 }$ , $B = \frac { 1 } { 2 }$ , $C = \frac { 1 } { 7 }$ , $D = \frac { 6 } { 7 }$

A market survey shows that 50% of the population used Brand Z computers last year, 8% of the population quit their jobs last year, and 4% of the population used Brand Z computers and then quit their jobs. Are the events of using Brand Z computers and quitting your job independent ?

According to a University of Maryland study of 200 samples of ground meats, the probability that one of the samples was contaminated by salmonella was 0.20. The probability that a salmonella-contaminated sample was contaminated by a strain resistant to at least one antibiotic was 0.82, and the probability that a salmonella-contaminated sample was contaminated by a strain resistant to at least three antibiotics was 0.55. Find the probability that a ground meat sample that was contaminated by an antibiotic-resistant strain was contaminated by a strain resistant to at least three antibiotics. Please round the answer to three decimal places. ​ The probability is __________.

Write down the transition matrix associated with the given state transition diagram below. $A = \frac { 1 } { 4 }$ , $B = \frac { 3 } { 4 }$ , $C = \frac { 1 } { 3 }$ , $D = \frac { 2 } { 3 }$

You are given a transition matrix P. Find the steady-state distribution vector. ​ $P = \left[ \begin{array} { l l } \frac { 1 } { 5 } & \frac { 4 } { 5 } \\\frac { 1 } { 3 } & \frac { 2 } { 3 }\end{array} \right]$

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places. $Y _ { 1 }$ , $Y _ { 2 }$ , $Y _ { 3 }$ form a partition of S. $P \left( X \mid Y _ { 1 } \right) = 0.2$ , $P \left( X \mid Y _ { 2 } \right) = 0.3$ , $P \left( X \mid Y _ { 3 } \right) = 0.6$ , $P \left( Y _ { 1 } \right) = 0.2$ , $P \left( Y _ { 2 } \right) = 0.4$ . Find $P \left( Y _ { 1 } \mid X \right)$ .

The Sorry State Lottery requires you to select 7 different numbers drawn at random from a set numbered 1 to 43. (Order is not important.) You are a Big Winner if the 7 numbers you select agree with those in the drawing. What is the probability of being a Big Winner

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 had a Math SAT of 700 or above or was admitted." (Round your answer to the nearest 0.01). - - - - - Total Admitted 6 212 1,127 2,882 5,211 9,438 Not Admitted 754 3,518 8,731 12,230 5,150 30,383 Total Applicants 760 3,730 9,858 15,112 10,361 39,821 ​ Enter your answer in decimal form, not as a fraction. The probability is __________.

The following table shows the results of a survey of 410 authors by a (fictitious) publishing company: New Authors Established Authors Total Successful 80 50 130 Unsuccessful 130 150 280 Total 210 200 410 ​ An author is chosen at random. Compute the estimated probability of the event that the author is a new author.

In 2003, the probability that a randomly chosen bag of cement from the world supply was consumed in the one country was 0.04, while the probability that it was consumed in another country was 0.50. What is the probability that a randomly chosen bag of cement was consumed in neither country ?

Find the conditional probability of the event "The sum is 5, given that the green one is either 2 or 3" when two fair dice (one red and one green) are rolled.

Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round the answer to four decimal places. ​ $Y _ { 1 }$ , $Y _ { 2 }$ , $Y _ { 3 }$ form a partition of S. ​ $P \left( X \mid Y _ { 1 } \right) = 0.4$ , $P \left( X \mid Y _ { 2 } \right) = 0.5$ , $P \left( X \mid Y _ { 3 } \right) = 0.6$ , $P \left( Y _ { 1 } \right) = 0.8$ , $P \left( Y _ { 2 } \right) = 0.1$ . ​ Find $P \left( Y _ { 1 } \mid X \right)$ . ​ $P \left( Y _ { 1 } \mid X \right) =$ __________

It rains in Spain an average of once in 20 days, and when it does, hurricanes have a 4% chance of happening in Hartford. When it does not rain in Spain, hurricanes have a 1% chance of happening in Hartford. What is the probability that it rains in Spain when hurricanes happen in Hartford Please, round your answer to four decimal places. The probability is __________.

Calculate the (theoretical) probability P(E), using the given information, assuming that all outcomes are equally likely. n ( S ) = 9 and n ( E) = 5

Find $P \left( A ^ { \prime } \right)$ if $P ( A ) = 0.67$ . ​ $P \left( A ^ { \prime } \right) =$ __________

The following table shows the results of a survey of 360 authors by a (fictitious) publishing company: New Authors Established Authors Total Successful 90 10 100 Unsuccessful 110 150 260 Total 200 160 360 An author is chosen at random. Compute the estimated probability of the event that the author is successful.

A bag contains 3 red marbles, 2 green ones, 1 fluorescent pink one, 2 yellow ones, and 2 orange ones. Suzy grabs 4 at random. Find the probability of the event: "She gets all the red ones, given that she does not get the fluorescent pink one."

A pair of dice is cast, and the number that appears uppermost on each die is observed. Find the probability of the event that the sum of the numbers is 9.