Consider a security with the stock prices
S(1) =



80 with probability 1/8
90 with probability 2/8
100 with probability 3/8
110 with probability 2/8
(a) What is the current price of the stock for which the expected return
would be 12%?
(b) What is the current price of the stock for which the standard deviation
would be 18%

Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1) complete parts (a) through (d)
a. The probability that Z is less than -1.59 is _______
b. The probability that Z is greater than 1.81 is________
c. The probability that Z is between -1.59 and 1.81 is______
d. The probability that Z is less than -1.59 or greater than 1.81______

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

1. What is ?(?=?2,?=?2)P(A=a2,B=b2)?

2. Observing events from this probability distribution, what is the probability of seeing (a1, b1) then (a2, b2)?

3. Calculate the marginal probability distribution, ?(?)P(A).

4. Calculate the marginal probability distribution, ?(?)P(B).

5.   According to a reliable source, 65% of murders are committed with a firearm. Suppose 15 murders are randomly selected. First construct a relative and cumulative frequency distribution for the situation. Then confirm that it is both a probability and binomial probability distribution.

a.   Compute the mean.
b.   Compute the standard deviation.
c.   Would a sample of 15 with 6 murders committed with a firearm be considered unusual? Justify your reasoning.
d.   Find the probability that exactly 10 murders are committed with a firearm.
e.   Find the probability that at most 11 murders are committed with a firearm.
f.   Find the probability that at least 12 murders are committed with a firearm
i.   Find the probability that between 9 and 13 murders are committed with a firearm.

Problem 4S-1

Consider the following system:

→ 0.74 → 0.74 →

Determine the probability that the system will operate under each of these conditions:

a. The system as shown. (Do not round your intermediate calculations. Round your final answer to 4 decimal places.)

Probability              

b. Each system component has a backup with a probability of .74 and a switch that is 100% percent reliable. (Do not round your intermediate calculations. Round your final answer to 4 decimal places.)

Probability            

c. Backups with .74 probability and a switch that is 99 percent reliable. (Do not round your intermediate calculations. Round your final answer to 4 decimal places.)

Probability            

Students taking the GMAT were asked about their undergraduate major and pursuit of their MBA as full time or part time student,

If a student taking the GMAT is randomly selected from this distribution find:

1. The probability that their undergraduate major was business.
2. The probability that their undergraduate major was not business.
3. The probability that their undergraduate major was business and they are a part time student.
4. The probability that their undergraduate major was business or they are a part time student.
5. The probability that they are in business given that they are a part time student.
6. The probability that they are a part time student given that they are in business.

Jeff is a sports fan. He has a wish list to see a 1. baseball, 2. basketball, and 3. football game this year. Two of his friends independently deicde to buy him tickets to one event. John has a probability of selecting tickets to 1,2, and 3 with a probability 1/5, 2/5, and 2/5. Jason has a probability of selecting tickets to 1, 2, and 3, respectively with probability 4/7, 1/7, 2/7, respectively. What is the probability John and Jason give Jeff a ticket to a different type of sporting event? Given the tickets are to different sporting events, what is the probability the event is baseball?

1. Assumptions: Two child family, the probability of a boy or girl is .5, sex of one child in the family is independent of the sex of the other child.

Case A: With no other information given, what is the probability that a family has 2 girls?

Case B: A family has at least 1 girl, what is the probability that a family has 2 girls?

Case C: A family has at least 1 girl who is its first born child, what is the probability that a family has 2 girls?

Comment: As we move from Case A to Case B to Case C, we have more information and the probability space shrinks and the probability of a 2-girl family increases.

an insurance company issues life insurance policies in three separate categories: standard,preferred,and ultra- preferred. Of the company's policyholders, 30%are standard,50% are preferred, and 20% are ultra-preferred. each standard policyholder has a probability 0.015 of dying in the next year, each preferred policyholder has probability 0.002 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year.

a) what is the probability that a policyholder has the ultra-preferred policy and dies in the next year?

b) what is the probability that a policyholder dies in the next year?

c) a policyholder dies in the next year. what is the probability that the deceased policyholder was ultra-preferred?

Given a normal distribution with μ = 100 and σ=10, complete parts​ (a) through​ (d).

a. What is the probability that X > 95?

The probability that X > 95 is ___.

​(Round to four decimal places as​ needed.)

b. What is the probability that X < 75​?

The probability that X < 75 is ___.

​(Round to four decimal places as​ needed.)

c. What is the probability that X < 85 or X > 110​?

The probability that X < 85 or X > 110 is___.

​(Round to four decimal places as​ needed.)

d. 90​% of the values are between what two​ X-values (symmetrically distributed around the​ mean)?

90​% of the values are greater than ____ and less than ____.

​(Round to four decimal places as​ needed.)