Probability and probability distributions are important concepts in applied business statistics. Most of the business decision making processes involve some uncertainty and randomness, which require the use of probability and probability distributions.

What is the importance of probability and probability distributions in a business decision making processes such as scenario analysis, sales forecasting and risk evaluation? Explain your responses by applying specific application examples of probability and probability distributions in business and discuss their implications on business management efficiency.

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, r_B, for Stock B (r_A = 15.20%.) Do not round intermediate calculations. Round your answer to two decimal places.
   %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 24.39%.) Do not round intermediate calculations. Round your answer to two decimal places.
   %

3. Now calculate the coefficient of variation for Stock B. Round your answer to two decimal places.

4. Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   2. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   4. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, r_B, for Stock B (r_A = 14.00%.) Do not round intermediate calculations. Round your answer to two decimal places.
   %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 22.91%.) Do not round intermediate calculations. Round your answer to two decimal places.
   %

3. Now calculate the coefficient of variation for Stock B. Round your answer to two decimal places.
   

Calculate the following binomial probability by either using one of the binomial probability tables, software, or a calculator using the formula below. Round your answer to 3 decimal places. P(x | n, p) = n! (n − x)! x! · px · qn − x where q = 1 − p P(x > 15, n = 20, p = 0.8) =

Calculate the following binomial probability by either using one of the binomial probability tables, software, or a calculator using the formula below. Round your answer to 3 decimal places. P(x | n, p) = n! (n − x)! x! · px · qn − x where q = 1 − p P(x = 11, n = 13, p = 0.80) =

Stocks A and B have the following probability distributions of expected future returns:

1. Calculate the expected rate of return, , for Stock B ( = 12.70%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 21.02%.) Do not round intermediate calculations. Round your answer to two decimal places.

     %

   Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense.
   2. If Stock B is more highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   4. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.

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3. Assume the risk-free rate is 3.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to four decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense.
   2. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   3. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
   4. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense.
   5. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.

Calculate the following binomial probability by either using one of the binomial probability tables, software, or a calculator using the formula below. Round your answer to 3 decimal places.

P(x | n, p) =

· p^x · q^{n − x}    where    q = 1 − p

P(x < 19, n = 20, p = 0.9)

=

(c) The probability of returns from an asset are as follows: Probability Return 35% 8% 55% 17% 10% -2% Determine the risk (i.e., standard deviation of returns) of the asset. [5 marks]

(d) Prepare a loan amortisation schedule for a term loan entailing $5,000 borrowed over 2 years at 8.5% p.a. interest repaid annually. [5 marks]

(e) For a rapidly growing Germancompany, the projectedgrowth rate is 10% for the next twoyears and 7% for the two yearsfollowing that. At the end of 4 years, the growth rate will expectantly settle to4% and remain so for the foreseeable future. The company expects to paya dividend of€2.5 per share next year. Assume that the investors' required rate of return for the company's shares is 20%. Determine the value of this company's share.Is the share a desirable purchase if its market price is £25€25per share? [7.5 marks

The probability destiny function is where statistics and probability come together. While there are several different kinds of discrete probability functions (or PDF's), three in particular are most commonly used. These are the binomial, Poisson and hypergeometric. What are the characteristics of each? Where and how are they used? Have you ever seen or even used any of these?