Your driving time to work T (continuous random variable) is between 22 and 64 minutes if the day is sunny, and between 41 and 86 minutes if the day is rainy, with a uniform probability density function in the given range in each case. Assume that a day is sunny with probability p subscript s = 0.41 and rainy with probability p subscript r equals 1 minus p subscript s. Your distance to work is X = 60 kilometers. Let V be your average speed for the drive to work, measured in kilometers per minute: V equals X over T Compute the value of the probability density function (PDF) of the average speed V at V = 0.62

A covid-19 test manufacturer is testing its newest instant test. It tests a total of 500 people, 100 of whom are known to be have the virus. The following table shows the results:

1. Find the probability that a person has the virus but tests negative.
2. Find the probability that a person who has the virus tests negative.
3. Find the probability that a person who tests negative has the virus.
4. An acquaintance of yours has a negative test result. What is the probability that she does not have the virus?

According to Masterfoods, the company that manufactures M&M’s, 12% of peanut M&M’s are brown, 15% are yellow, 12% are red, 23% are blue, 23% are orange and 15% are green. You randomly select five peanut M&M’s from an extra-large bag of the candies. (Round all probabilities below to four decimal places; i.e. your answer should look like 0.1234, not 0.1234444 or 12.34%.)

Compute the probability that exactly four of the five M&M’s are orange.

Compute the probability that three or four of the five M&M’s are orange.

Compute the probability that at most four of the five M&M’s are orange.

Compute the probability that at least four of the five M&M’s are orange.

In a web homework problem, we might generate a system of equations ax + by = k 1 , cx + dy = k 2 , where
each of a, b, c, d, k 1 and k 2 are randomly generated. Suppose that each of these values is equally likely
to be 1, 2 or 3, and that they are generated independently.
a) Let the random variable X = ad. What are possible values of X? What is its probability mass
function?
b) What is the probability that the system may not have a solution—that is, what is the probability
that ad = bc?
c) Now suppose that the system is not guaranteed a solution (that is, we know that ad = bc). What is
the probability that bc has one of the values 2, 3, or 6?

Motorcycle riders wear high visibility clothing so that drivers can spot them easily. 50% of Motorcyclist wear high visibility clothing. A researcher experiments and determines that if drivers see a Motorcyclist, then the probability the Motorcyclist was wearing high visibility clothing was 0.51. That experiment shows that 92% of Motorcyclist were seen by drivers on the road.

A. Find the probability that a driver does not see a Motorcyclist , give that the Motorcyclist was wearing visibility clothing?

B. Find the probability that a driver does not see a Motorcyclist , given that the Motorcyclist was not wearing high visibility clothing

C. If 200 drivers pass a Motorcyclist wearing high visibility clothes, what's the probability that at least one driver doesn't see the Motorcyclist.

A survey on ownership of satellite TV systems shows the following data:

                                                            Region

Complete the table and answer the following: (Write answers either as fractions or as decimals to 4 decimal places.)

a) What is the probability that someone is likely to purchase a satellite system?

b) What is the probability that someone owns a system and lives in the southwest?

c) Given a person lives in the west, what is the probability that they are likely to purchase a satellite system?

d) Given that someone is likely to purchase a system, what is the probability that they live in the east?

Cash flow series

Determine the expected present worth of the following cash flow series if each series may be realized with the probability shown at the head of each column. Let i = 20% per year.

The present worth when the probability is 0.3 is $

The present worth when the probability is 0.22 is $

The present worth when the probability is 0.48 is $

The expected present worth value is $  .

Problem 3

Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of its being clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that these probabilities do not change if information is also provided about the weather before today.

a) Find the n-step transition matrix for n = 3, 7, 10, 20.

b) The probability of rain today is 0.5 Use the results in the part (a) to determine the probability that it rains in n days, for n = 3, 7, 10, 20.

c) Formulate the steady-state equations to obtain the steady state probabilities for this process. Compare the probabilities of the transition matrices of n steps of part (a) with these of the steady state as n grows.

In a deck of 52 playing cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K in each four suits),                     

1. What is the probability of selecting a black (i.e., club or spade) king card?
2. Without replacement means the card IS NOT put back into the deck. What is the probability that when two cards are drawn from a deck of cards without replacement that both of them will be 8’s?
3. What is the probability of being dealt a flush (5 cards of all the same suit) from the first 5 cards in a deck?
4. With replacement means the card IS put back into the deck. What is the probability of drawing an Ace 3 times in a row with replacement?

PLEASE SHOW STEPS AND EXPLAIN

A biologist needs at least 4 mature specimens of a certain plant. The plant needs a year to
reach maturity; once a seed is planted, any plant will survive for the year with probability
1/1000 (Independently of other plants). The biologist plants 2000 seeds. A year is deemed a
success if 4 or more plants from these seeds reach maturity.
a. Write down the exact expression for the probability that the biologist will indeed end up
with at least 4 mature plants.
b. Write down the relevant approximate expression for the probability from (a). Justify
briefly the approximation.
c. The biologist plans to do this year after year. What is the approximate probability that he
has at least 3 success in 8 years?