Question 1

Although he is very poor,Al plays the million-dollar lottery every day because he is certain that one day he will win.Al makes this calculation based upon

Accepted Answer

Al is using subjective probability, which is based on personal belief and perception rather than actual frequency or knowledge of all possible outcomes. He is certain that he will win, despite the low odds of winning, because he has a strong personal feeling or belief about it.

Question 2

Expected value represents the average of all outcomes if one were to undertake the risky event many times over and over again.

Accepted Answer

The expected value is not expected on any one outcome,because it represents the average of many outcomes.

Question 3

Sarah buys little stuffed animals for $5 each.They come in different varieties.If the producer stops making (retires)a certain variety,a stuffed animal of that variety will be worth $100; otherwise it is worth $0.There is 50% chance that any variety will be retired.What is the value to Sarah of knowing ahead of time whether a variety will be retired?

Accepted Answer

The value of knowing ahead of time is calculated as the difference between the expected value with and without the information. Without the information, the expected value of a stuffed animal is 0.5 * $100 + 0.5 * $0 = $50. With the information, the value is either $100 (if retired) or $0 (if not), but since she would only buy the retired ones, her expected value is $100. The value of the information is $100 - $50 = $50. However, since she pays $5 for each stuffed animal, the net value to Sarah of knowing ahead of time is $50 - $5 = $45. This option is not listed, indicating a misunderstanding in the explanation process. The correct interpretation should focus on the expected gain from the decision to buy or not buy based on the information, which is the increase in expected value due to having the information. Given the options, the closest correct approach to understanding the value of the information is by considering how it affects the decision-making process in terms of expected outcomes, which seems to have been misinterpreted in the initial explanation.

Question 4

Explain why the variance of an investment is a useful measure of the risk associated with it.

Accepted Answer

The answer of Explain why the variance of an investment...

Question 5

If there are 10,000 people in your age bracket,and 10 of them died last year,an insurance company believes that the probability of someone in that age bracket dying this year would be

Accepted Answer

The probability of someone in the age bracket dying this year is the number of people who died last year divided by the total number of people in the age bracket.

Probability = Number of deaths / Total number of people

So,

Probability = 10 / 10,000 = 0.001

Therefore, the answer is B, which is 0.001.

Question 6

On any given day we know a salesman can earn $0 with a 30% probability,$100 with a 20% probability or $300 with 40% probability.His expected earnings equal

Accepted Answer

Expected earnings can be calculated by multiplying the probability of each outcome by its corresponding value and then adding them up. 
Expected Earnings = (0.3 x 0) + (0.2 x 100) + (0.4 x 300) 
Expected Earnings = 0 + 20 + 120 
Expected Earnings = $140

Question 7

All else held constant,as the variance of a payoff increases,the

Accepted Answer

Variance measures the spread of the possible outcomes, so as the variance of a payoff increases, the range of potential outcomes widens, increasing the level of risk involved.

Question 8

If a payout is certain to occur,then the variance of that payout equals

Accepted Answer

If the payout is certain to occur, then there is no uncertainty or variability in the payout, meaning there is no variance. The formula for variance includes squaring the difference between each possible outcome and the expected value, but since there is only one possible outcome (the certain payout), there is nothing to square or calculate. Therefore, the variance would be zero.

Question 9

On any given day,a salesman can earn $0 with a 20% probability,$100 with a 40% probability,or $300 with a 20% probability.Calculate the expected value and variance of his earnings,and interpret.

Accepted Answer

The answer of On any given day,a salesman can earn...

Question 10

On any given day,a salesman can earn $0 with a 30% probability,$100 with a 20% probability,or $300 with a 50% probability.His expected earnings equal

Accepted Answer

Expected earnings = (Probability of earning $0 * $0) + (Probability of earning $100 * $100) + (Probability of earning $300 * $300) 
= (0.3 * 0) + (0.2 * 100) + (0.5 * 300) 
= 0 + 20 + 150 
= $170

Therefore, the best choice would be option D with an expected earnings of $170.

Question 11

Sarah buys little stuffed animals for $5 each.They come in different varieties.If the producer stops making (retires)a certain variety,a stuffed animal of that variety will be worth $100; otherwise it is worth $0.There is 50% chance that any variety will be retired.When Sarah buys her next stuffed animal,the expected profit is

Accepted Answer

The answer of Sarah buys little stuffed animals for $5...

Question 12

A lottery game pays $500 with .001 probability and $0 otherwise.The variance of the payout is

Accepted Answer

The variance of a discrete probability distribution can be found using the formula:
Var(X) = Σ [ (xi - μ)^2 * P(xi) ]
where xi is the payout for each possible outcome, μ is the expected payout, and P(xi) is the probability of each outcome.

In this case, we have two possible outcomes:
- $500 with probability 0.001
- $0 with probability 0.999

So we can calculate the expected payout as:
E(X) = $500 * 0.001 + $0 * 0.999 = $0.50

Using the formula for variance, we get:
Var(X) = (500 - 0.5)^2 * 0.001 + (0 - 0.5)^2 * 0.999
Var(X) = 249.75

Therefore, the best choice is C.

Question 13

For a given expected value,the smaller the standard deviation of the expected value,the larger the risk.

Accepted Answer

The smaller the standard deviation,the smaller the associated risk.The standard deviation is the square root of the variance,thus the smaller the variance,the smaller the standard deviation.The smaller the variance of the expected value,the smaller the risk.

Question 14

Your friend Diana tells you that she thinks that her favorite softball team has a 70% chance of winning the next game because that is exactly the winning rate of her team in the last two seasons.This is an example of a(n)

Accepted Answer

The answer of Your friend Diana tells you that she...

Question 15

On any given day,a salesman can earn $0 with a 20% probability,$100 with a 40% probability,or $300 with a 20% probability.His expected earnings equal

Accepted Answer

The expected earnings are calculated by multiplying each outcome by its probability and summing these products: (0 * 0.20) + (100 * 0.40) + (300 * 0.20) = 0 + 40 + 60 = $100.

Question 16

You draw colored balls out of a bag.You draw a red ball 30% of the time and a blue ball 70% of the time.For each draw,the blue outcome and the red outcome are

Accepted Answer

Mutually exclusive events are those that cannot happen at the same time. In this case, drawing a red ball and drawing a blue ball cannot happen at the same time, so the events are mutually exclusive. 
Exhaustive events are those that cover all possible outcomes. In this case, drawing a red ball or drawing a blue ball covers all possible outcomes, so the events are exhaustive. Therefore, both A and B are correct.

Question 17

Your friend Dimitre tells you that he thinks that his favorite basketball team has a 70% chance of winning the next game.This is an example of a(n)

Accepted Answer

Dimitre's statement is based on his personal belief or opinion, rather than on historical data or mathematical calculation. This type of probability is known as subjective probability.

Question 18

Expected value represents

Accepted Answer

Expected value is the average of all payments or outcomes that would be received if the risky event were undertaken many times. It takes into account all possible outcomes and their probabilities, not just the most likely or actual payment.

Question 19

Assume the following.In location A yearly temperatures range from -30°F to 100°F and in location B yearly temperatures range from 55°F to 75°F.In both locations the average yearly temperature equals 65°F.We can conclude that

Accepted Answer

Variance measures the spread of a set of numbers. Location A, with a wider range of temperatures from -30°F to 100°F, exhibits a greater spread compared to Location B's narrower range of 55°F to 75°F. Therefore, Location A has a higher variance, and since standard deviation is the square root of variance, it also implies a higher standard deviation for Location A.

Question 20

People in a certain group have a 0.3% chance of dying this year.If a person in this group buys a life insurance policy for $3,300 that pays $1,000,000 to her family if she dies this year and $0 otherwise,what is the expected value of a policy to the insurance company?

Accepted Answer

To find the expected value, we multiply the probability of the event (in this case the person dying) by the payoff.

The probability of the person dying is 0.3%, which is equal to 0.003 (0.3/100).

The payoff to the insurance company is the premium they receive, which in this case is $3,300, minus the expected payout, which is equal to 0.003 x $1,000,000 = $3,000.

Therefore, the expected value to the insurance company is $300, which is the premium minus the expected payout.

Hence, the best choice is B.

Quiz 17: Uncertainty