1. A computer is inspected at the end of every hour. It is found to be either working (up) or failed (down). If the computer is found to be up, the probability of its remaining up for the next hour is 0.95. If it is down, the computer is repaired, which may require more than 1 hour. Whenever the computer is down (regardless of how long it has been down), the probability of its still being down 1 hour later is 0.5. The (one-step) transition matrix for this Markov chain is given by P = [0.95 0.05]. 0.5 (a) find the µ₁j (the expected first passage time from state į to state j) for all i and j. (b) Use the results from part (a), to answer the following questions. What is the expected number of hours that the computer will be keep working? What is the expected number of hours that the computer will be working again after it fails?

66% of college biology students in the United States are female. If we take a random sample of 200 of these students, find the probability that the number of females is at least 135.Write your answer in decimal to four decimal places.

Based on a study from the Chronicles of Flippin'' Awesomeness, the probability that Napoleon and Pedro make it to their first period class on time is 0.45.  The probability that they make it to their first period class on time, given that they catch the bus is 0.71.  The probability that Napoleon and Pedro catch the bus and make it to their first period class on time is 0.50.  What is the probability that Napoleon and Pedro catch the bus?

Suppose it has been determined that the probability is 0.8 that a rat injected with cancerous cells will live. If 95 rats are injected, how many would be expected to die?

According to TMUSS Quarterly (Totally Made-Up Sports Statistics) April 2017, the probability that the Tampa Bay Buccaneers defense will have 3 or more sacks in the game is 0.05.  The probability that the Bucs will both score a touchdown on their opening drive and have 3 or more sacks in the game is 0.25.  The is the probability that they will either score a touchdown on the opening drive or have 3 or more sacks in the game is 0.25.  What is the probability that the Bucs will score a touchdown on their first drive? Answer in decimal form.  Round to two decimal places as needed.   Your Answer:

Based on a study from the Chronicles of Flippin'' Awesomeness, the probability that Napoleon and Pedro make it to their first period class on time is 0.29.  The probability that Napoleon and Pedro catch the bus is 0.87. However, the probability that they make it to their first period class on time, given that they catch the bus is 0.81.  What is the probability that Napoleon and Pedro catch the bus and make it to their first period class on time?

Please provide steps for how you got the solution to the problem provided below. A machine is used to produce one of each of two types of product (A and B) on alternating days, i.e., if A is produced today then B will be produced tomorrow and vice versa. Each day, there is a probability that the machine malfunctions. Malfunction probability after a day of operation is 0.03 when producing A and 0.07 when producing B. Once the machine malfunctions, it is no longer operable and will not be replaced until sometime in the future (beyond modeling horizon). Model the system as a Markov chain (give transition matrix on your paper). Suppose that on Monday product A was produced. Find the probability that the machine is still operable by the morning of Thursday. Your model should have a state ”malfunction”, which cannot transition to any other state other than itself.

The probability that Mr. Johnson goes for an evening walk on any given day during the winter is 0.56.  Out of 89 days of winter, the number of days that Mr. Johnson is expected not to take an evening walk to the nearest day, is

Suppose that it rains in Spain an average of once every 15 days, and when it does, hurricanes have a 5% chance of happening in Hartford. When it does not rain in Spain, hurricanes have a 3% chance of happening in Hartford. What is the probability that it rains in Spain when hurricanes happen in Hartford? (Round your answer to four decimal places.)