Whatever be the condition of the economy, the financial markets have been improving very quickly. An investor with the
ability to buy assets is looking at a portfolio and finds that the probability of stock A increasing in value is 50.6%. She also
finds that if stock A increases in value then there is a 55% chance that stock B will also increase. However, if Stock A does
not increase in value, then B increases in only 25% of the times.
a) (3) Find the probability that both A and B increase in value.

b) (3) Find the probability of B increasing in value.

c) (3) Find the probability that either A or B or both increase in value.

d) (3) Find the probability that A increases in value, given that B increased.

e) (2) Are these stocks mutually exclusive? Show evidence to support your answer.

f) (2) Are these stocks statistically independent? Show evidence to support your answer.

g) (2) Are these stocks collectively exhaustive? Show evidence to support your answer.

Thousands of college students are attending a political conference and are about to enjoya dinnerbanquet. The probability a randomly selected student is a Democrat is .70. The probability a randomly selected student is a Republican is .30.Before the banquet, students were asked what they wanted for dinner. Their options were a vegetarian dish, beef, or chicken.Among the Democrat students, the proportion who requested the vegetarian dish was .20, the proportion who requested beef was .30, and the proportion who requested chicken was .50.Among the Republican students, the proportion who requested the vegetarian dish was .10, the proportion who requested beef was .50, and the proportion who requested chicken was .40.Draw a tree diagram and use it to answer these questions:3aIf a student is selected at random, what is the probability they are both a Democrat and requested chicken for dinner?3bIf a student is selected at random, what is the probability they requested the vegetarian dish for dinner?3cGiven that a randomly selected student requested beef for dinner, what is the probability they are a Republican?

The test scores for a math exam have a mean of 72 with a standard deviation of 8.5.

Let the random variable X represent an exam score.

a) Find the probability that an exam score is at most 80. (decimal answer, round to 3 decimal places)

b) Find the probability that an exam score is at least 60. (decimal answer, round to 3 decimal places)

c) Find the probability that an exam score is between 70 and 90. (decimal answer, round to 3 decimal places)   

d) Find the 75th percentile score. (round to 1 decimal place)

e) Would it be unusual for a random exam to have a score of 92?

A) Yes, because the probability of scoring 92 is below 0.05.

B) Yes, because a score of 92 is more than 2 standard deviations above the mean.

C) No, because the probability of scoring 92 is above 0.05.

D) No, because a score of 92 is within 2 standard deviations of the mean.

E) None of the above.

The Masterfoods Company says that yellow candies make up 20% of their plain M&M’s, red another 20%, and orange, blue, and green each make up 10%. The rest are brown.

a. If you randomly pick two M&M’s, what’s the probability that they are both brown? Round to TWO digits beyond the decimal.

b. If you randomly pick two M&M’s, what’s the probability that exactly one of the two is yellow? Round to TWO digits beyond the decimal.

c. If you randomly pick three M&M’s, what’s the probability that the third one is the first one that’s red?Round to THREE digits beyond the decimal.

d. If you randomly pick three M&M’s, what’s the probability that none are yellow? Round to THREE digits beyond the decimal.

e. If you randomly pick four​ M&M’s, what’s the probability that at least one is ​green? Round to THREE digits beyond the decimal.

The National Vaccine Information Center estimates that 91.2% of Americans have had chickenpox by the time they reach adulthood.

(a) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood.

(b) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood?

(c) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox?

(d) What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox? We now consider a random sample of 120 American adults.

(e) How many people in this sample would you expect to have had chickenpox in their childhood? And with what standard deviation?

(f) What is the probability that 96 or fewer people in this sample have had chickenpox in their childhood? (round to 4 places)

The National Vaccine Information Center estimates that 88.5% of Americans have had chickenpox by the time they reach adulthood.
(a) Calculate the probability that exactly 98 out of 100 randomly sampled American adults had chickenpox during childhood.

(b) What is the probability that exactly 2 out of a new sample of 100 American adults have not had chickenpox in their childhood?

(c) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox?

(d) What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox?

We now consider a random sample of 120 American adults.
(e) How many people in this sample would you expect to have had chickenpox in their childhood?

And with what standard deviation?

(f) What is the probability that 96 or fewer people in this sample have had chickenpox in their childhood? (round to 4 places)

During the embryogenesis, axial mesoderm cells are specialized to express different genes. We identified three group of cells present in close proximity. Among these groups one cell subtype (known as Sub1) have probability of 0.5 to express gene XRA. The other cell subtypes (known as Sub2 and Sub3) have a probability of 0.8 and 0.1 respectively.

Suppose that we isolated one of these cell subtypes at random and grow them in the lab to study gene expression patterns. We define two states for gene expression; as the state of success (S) if gene XRA is observed and failure (F) if not.

a) What is the probability (P(S)) of observing gene XRA expression?

b) Given that gene XRA will be expressed every hour. What is probability P(SFS) in the next three hours?

c) Assuming that we observed the sequence of SFF, what is the probability of that we isolated the subtype 2 cells during the embryogenesis?

Pr9.

Suppose the emergency room at Mass General opens at 6am and has a mean arrival rate throughout the day of 6.9 patients per hour (that is λ = 6.9).

(A) What is the probability that 12 patients arrive between 6am and 7am?

(B) What is the probability that no patient arrives before 7am?

(C) What is the probability that the first patient arrives between 6am and 7am?

(D) What is the probability that the first patient arrives between 6:15 and 6:45?

(E) Suppose it is 6:15 and no patient has arrived yet; now what is the probability that the first patient arrives between 6:15 and 6:45?

Hint: Use the Poisson for (A) and (B) and the Exponential for (C), (D), and (E). Note carefully how (D) and (E) are different: in (D) it is possible that the first patient arrives between 6am and 6:15am, whereas for (E) you know this has not happened. The result for the two will be different!

The overhead reach distances of adult females are normally distributed with a mean of 205.5 cm and a standard deviation of 8 cm. a. Find the probability that an individual distance is greater than 218.90 cm. b. Find the probability that the mean for 25 randomly selected distances is greater than 203.30 cm. c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

a. The probability is nothing. ​(Round to four decimal places as​ needed.)

b. The probability is nothing. ​(Round to four decimal places as​ needed.)

c. Choose the correct answer below.

A. The normal distribution can be used because the probability is less than 0.5

B. The normal distribution can be used because the finite population correction factor is small.

C. The normal distribution can be used because the original population has a normal distribution.

D. The normal distribution can be used because the mean is large.

what are the formulas for each column so i can understand where the numbers are coming from. refer to Question: Historical data indicate that a student's income for any month...part of the answer has been posted on Chegg but the formulas are needed.Thanks!

Assuming the student begins the school year with a balance of $1200, use Excel to simulate 12 months of activity and to predict the position of the student at the end of the year.

Historical data indicate that a student’s income for any month of school from work, parents, scholarships, and loans is consistent with the following probability distribution: