You are analyzing stocks of three companies you are interested in investing in. Suppose that these three stocks are Apple stock (AAPL), Microsoft (MSFT), and Tesla (TSLA). At the end of this month, each month may move up (increase in value), move down (decrease in value) or stay the same. In this case, an experiment consists of observing the price condition across the three stocks. Each of the three conditions (move up, move down, stay same) are equally likely.

How many outcomes are in the sample space?

List 7 of the outcomes. (use U for move up, D for moving down and S for staying the same)

Let A be the event that all the stocks have the same performance. List the outcomes in A.

Let B be the event that all of stocks are different. List the outcomes in B.

Let C be the event that at least two stocks move up.

Are events A and C mutually exclusive?

Are events B and C mutually exclusive?

What is the probability of ?

What is the probability of ?

What is the probability of ?

What is the probability of ?

Suppose that on any given day, there is a 20% chance that Lebron has a bad day, a 50% day that Lebron has a so-so day, and a 30% chance that Lebron has a good day. On a bad day, the probability that Lebron makes any given 3-point shot is 0.5. On a so-so day, the probability that Lebron makes any given 3-point shot is 0.7. On a good day, the probability that Lebron makes any given 3-point shot is 0.9. Suppose that tomorrow, Lebron takes a 3-point shot, and we record whether he hits or misses and what kind of day he was having.

What is the probability that Lebron hits?

What is the probability that if Lebron is having a bad day given that he misses a shot that day?

Suppose that a local factory uses a type of equipment called a super-drill. Suppose that at the start of each day, the machine must be restarted. Suppose that upon restarting, there is a .95 probability that the machine boots into normal mode, and a .05 probability that the machine boots into problem mode, independent of all else. Suppose that if the machine is in normal mode, each hole drilled is high quality with probability .98, and low quality with probability .02, independently for each hole drilled that day. Suppose that if the machine is in problem mode, each hole drilled is high quality with probability .55, and low quality with probability .45, independently for each hole drilled that day.

What is the probability that the first hole drilled tomorrow is of low quality?

Suppose that you are told that the first hole drilled tomorrow is of high quality. What is the conditional probability that the drill is in normal mode that day?

Suppose that you are told that tomorrow exactly 10 holes are drilled, and that at least 2 of these holes are high quality. What is the conditional probability that the drill is in normal mode that day?

A local coffee shop in Greensboro sells coffee cups. Suppose that coffee cups are only one size and one type. The cup of coffee sells for $2.50 per cup. The daily demand for coffee (from that local shop in cups) can be expressed by the following probability mass function.

x

f(x)

15

0.06

20

0.22

25

0.37

30

0.25

35

0.05

45

0.05

The store manager estimates the cost of a cup of coffee to be $1.25 per cup. If she places an order at the beginning of the day for 30 units,

How much time would she expect to sell?

How much profit would she expected to make?

How many cups would she expect to have left at the end of the day?

A quality control engineer samples five from a large lot of manufactured firing pins and checks for defects. Unknown to the inspector, three of the five sampled firing pins are defective. The engineer will test the five pins in a randomly selected order until a defective is observed (in which case the entire lot will be rejected). Let Y be the number of firing pins the quality control engineer must test.

a) Find the probability distribution of Y.

b) Suppose the cost of testing a single firing pin is $200. What is the expected cost of inspecting
the lot?

Pedestrians arrive at a street crossing according to a Poisson process at a rate of 4 per minute. A traffic light alternates between two states with durations shown below:

Don’t Walk 2 minutes

Walk 1 minute

What is the mean and standard deviation of the number of pedestrians waiting to cross?

What is the probability that nobody is waiting to cross?

What is the probability that nobody crosses during the entire one minute Walk cycle?

The life of a certain type of device has an exponentially distributed failure with rate .008 per day.
a) What is the mean time to failure?

b) What is the probability that 200 days will pass before a failure is observed?

c) What is the probability that 4 of the next 6 failures observed will occur after 200 days have
passed?

d) What is the probability that at most 3 failures will occur in 200 days?

Question

You are analyzing stocks of three companies you are interested in investing in. Suppose that these three stocks are Apple stock (AAPL), Microsoft (MSFT), and Tesla (TSLA). At the end of this month, each month may move up (increase in value), move down (decrease in value) or stay the same. In this case, an experiment consists of observing the price condition across the three stocks. Each of the three conditions (move up, move down, stay same) are equally likely.
How many outcomes are in the sample space?
List 7 of the outcomes. (use U for move up, D for moving down and S for staying the same)
Let A be the event that all the stocks have the same performance. List the outcomes in A.
Let B be the event that all of stocks are different. List the outcomes in B.
Let C be the event that at least two stocks move up.
Are events A and C mutually exclusive?
Are events B and C mutually exclusive?
What is the probability of ?
What is the probability of ?
What is the probability of ?
What is the probability of ?

Suppose that on any given day, there is a 20% chance that Lebron has a bad day, a 50% day that Lebron has a so-so day, and a 30% chance that Lebron has a good day. On a bad day, the probability that Lebron makes any given 3-point shot is 0.5. On a so-so day, the probability that Lebron makes any given 3-point shot is 0.7. On a good day, the probability that Lebron makes any given 3-point shot is 0.9. Suppose that tomorrow, Lebron takes a 3-point shot, and we record whether he hits or misses and what kind of day he was having.

What is the probability that Lebron hits?
What is the probability that if Lebron is having a bad day given that he misses a shot that day?

Suppose that a local factory uses a type of equipment called a super-drill. Suppose that at the start of each day, the machine must be restarted. Suppose that upon restarting, there is a .95 probability that the machine boots into normal mode, and a .05 probability that the machine boots into problem mode, independent of all else. Suppose that if the machine is in normal mode, each hole drilled is high quality with probability .98, and low quality with probability .02, independently for each hole drilled that day. Suppose that if the machine is in problem mode, each hole drilled is high quality with probability .55, and low quality with probability .45, independently for each hole drilled that day.
What is the probability that the first hole drilled tomorrow is of low quality?
Suppose that you are told that the first hole drilled tomorrow is of high quality. What is the conditional probability that the drill is in normal mode that day?
Suppose that you are told that tomorrow exactly 10 holes are drilled, and that at least 2 of these holes are high quality. What is the conditional probability that the drill is in normal mode that day?

A local coffee shop in Greensboro sells coffee cups. Suppose that coffee cups are only one size and one type. The cup of coffee sells for $2.50 per cup. The daily demand for coffee (from that local shop in cups) can be expressed by the following probability mass function.

x	f(x)
15	0.06
20	0.22
25	0.37
30	0.25
35	0.05
45	0.05

The store manager estimates the cost of a cup of coffee to be $1.25 per cup. If she places an order at the beginning of the day for 30 units,

How much time would she expect to sell?
How much profit would she expected to make?
How many cups would she expect to have left at the end of the day?

A quality control engineer samples five from a large lot of manufactured firing pins and checks for defects. Unknown to the inspector, three of the five sampled firing pins are defective. The engineer will test the five pins in a randomly selected order until a defective is observed (in which case the entire lot will be rejected). Let Y be the number of firing pins the quality control engineer must test.
a) Find the probability distribution of Y.
b) Suppose the cost of testing a single firing pin is $200. What is the expected cost of inspecting
the lot?

Pedestrians arrive at a street crossing according to a Poisson process at a rate of 4 per minute. A traffic light alternates between two states with durations shown below:

Don’t Walk 2 minutes

Walk 1 minute

What is the mean and standard deviation of the number of pedestrians waiting to cross?
What is the probability that nobody is waiting to cross?
What is the probability that nobody crosses during the entire one minute Walk cycle?

The life of a certain type of device has an exponentially distributed failure with rate .008 per day.
a) What is the mean time to failure?
b) What is the probability that 200 days will pass before a failure is observed?
c) What is the probability that 4 of the next 6 failures observed will occur after 200 days have
passed?
d) What is the probability that at most 3 failures will occur in 200 days?

Accepted Answer

Answered: Image /qna-images/answer/1a8be9e3-78e3-4147-882b-55af1fb0cee1.jpg