Sally and Mike are playing frisbee at the beach. When Sally throws the frisbee the probability is 0.14 that it comes back to Sally, the probability is 0.7 that it goes to Mike, and the probability is 0.16 that the dog runs away with the frisbee. When Mike throws the frisbee there is a 0.66 probability that Sally gets it, a 0.28 probability that it comes back to Mike, and a 0.06 probability that the dog runs away with the frisbee. Treat this as a 3--state Markov Chain with the dog being an absorbing state. If Mike has the frisbee, what is the expected value for the number of times Mike will throw the frisbee before the dog gets it? (Give your answer correct to 2 decimal places.)

Complete parts​ (a) and​ (b) below. The number of dogs per household in a small town

Dogs 0 1 2 3 4 5

Probability 0.649 0.220 0.087 0.024 0.013 0.008 ​

(a) Find the​ mean, variance, and standard deviation of the probability distribution.

2. Students sometimes delay doing laundry until they finish their problem sets. Assume all random values described below are mutually independent.

(a) A busy student must complete 3 problem sets before doing laundry. Each problem set requires 1 day with probability 2/3 and 2 days with probability 1/3. Let B be the number of days a busy student delays laundry. What is E(B)?

Example: If the first problem set requires 1 day and the second and third problem sets each require 2 days, then the student delays forB = 5 days.

(b) A relaxed student rolls a fair, 6-sided die in the morning. If he rolls a 1, then he does his laundry immediately (with zero days of delay). Otherwise, he delays for one day and repeats the experiment the following morning. Let R be the number of days a relaxed student delays laundry. What is E(R)?

Example: If the student rolls a 2 the first morning, a 5 the second morning, and a 1 the third morning, then he delays for R = 2 days.

(c) Before doing laundry, a nostalgic student must dream of riding the LX bus for a number of days equal to the product of the numbers rolled on two fair, 6-sided dice. Let N be the expected number of days a nostalgic student delays laundry. What is E(N)?

Example: If the rolls are 5 and 3, then the student delays for N = 15 days.

(d) A student is busy with probability 1/2, relaxed with probability 1/3, and nostalgic with probability 1/6. Let D be the number of days the student delays laundry. What is E(D)?

If telephone purchase orders are Poisson distributed with an average of 3 orders per minute, answer the following:

1. probability at most 3 in a 30 second interval

2. probability of at least 4 in a 45 second interval

3. probability of 1, 2, or 3, in a 60 second interval

4. probability that of 5 different 45 second intervals 2 or more of the intervals would entail at least 4 orders each

5. standard deviation of the expected number of failures in a 60 second interval

According to an​ airline, flights on a certain route are on time 85​% of the time. Suppose 10 flights are randomly selected and the number of​ on-time flights is recorded. ​(a) Explain why this is a binomial experiment. ​(b) Find and interpret the probability that exactly 7 flights are on time. ​(c) Find and interpret the probability that fewer than 7 flights are on time. ​(d) Find and interpret the probability that at least 7 flights are on time. ​(e) Find and interpret the probability that between 5 and 7 ​flights, inclusive, are on time.

According to an​ airline, flights on a certain route are on time 75​% of the time. Suppose 10 flights are randomly selected and the number of​ on-time flights is recorded.

​(a) Explain why this is a binomial experiment.

​(b) Find and interpret the probability that exactly 6 flights are on time.

​(c) Find and interpret the probability that fewer than 6 flights are on time.

​(d) Find and interpret the probability that at least 6 flights are on time.

​(e) Find and interpret the probability that between 4 and 6 ​flights, inclusive, are on time.

A can company reports that the number of breakdowns per 8-hour shift on its machine operated assembly line follows a Poisson distribution with a mean of 1.5.

1. What is the probability of exactly three breakdowns on the midnight shift?
2. What is the probability of greater than two breakdowns in one hour?
3. What is the probability that the time between two successive breakdowns is at most three hours?
4. If two hours have gone by without a breakdown, what is the probability that a breakdown will occur in the next one hour?

According to an​ airline, flights on a certain route are on time 80​% of the time. Suppose 11 flights are randomly selected and the number of​ on-time flights is recorded. ​ ​(a) Determine the values of n and p. ​(b) Find and interpret the probability that exactly 9 flights are on time. ​(c) Find and interpret the probability that fewer than 9 flights are on time. ​(d) Find and interpret the probability that at least 9 flights are on time. ​(e) Find and interpret the probability that between 7 and 9 ​flights, inclusive, are on time.

You roll two fair dice. Let A be the event that the sum of the dice is an even number. Let B be the event that the two results are different.

(a) Given B has occurred, what is the probability A has also occurred?

(b) Given A has occurred, what is the probability B has also occurred?

(c) What is the probability of getting a sum of 9?

(d) Given that the sum of the pair of dice is 9 or larger, find the probability that the sum of the pair of dice is exactly 10.