Expected Value and Random Variables

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Questions and Answers

When a variable can take on different values, it is a:

Random variable (correct)

Dependent variable

Exogenous variable

Independent variable

The expected value is:

The product of the sums of the probability and the values in different states

The sum of the products of the probability and the values in different states (correct)

The difference between the products of the probability and the values in different states

The difference between the sums of the probability and the values in different states

Heads and tails are equally likely, but you win a dollar on heads and lose a dollar on tails. The expected value is:

$0.50

$1.00

$0.00 (correct)

-$0.50

Heads and tails are equally likely, but you win $2.00 on heads and lose $1.00 on tails. The expected value is:

$0.50 Signup and view all the answers

Heads and tails are equally likely, but you win $3.00 on heads and lose $1.00 on tails. The expected value is:

$1.00 Signup and view all the answers

You can invest $100,000 into either project A or B. You estimate that A succeeds with probability 0.7, doubling in value if it succeeds. Project B succeeds with probability 0.6. You should invest in:

Project A Signup and view all the answers

You can invest $100,000 into either project A or B. You estimate that A succeeds with probability 0.6, while project B succeeds with probability 0.7. You should invest in:

Project A Signup and view all the answers

You can invest $100,000 into either project A or B. You estimate that A succeeds with probability 0.5 while project B succeeds with probability 0.8. You should invest in:

Cannot tell from the information presented Signup and view all the answers

You can invest in either project A or B. Project A has value $100 with probability 0.1 and value $75 with probability 0.9. Project B has value $110 with probability 0.2 and value $70 with probability 0.8. You should invest in:

Cannot tell from the information presented Signup and view all the answers

Three possibilities are equally likely with payoffs of $3, $6, and $9. The expected value is:

$6 Signup and view all the answers

Four possibilities are equally likely with payoffs of $2, $4, $6, and $8. The expected value is:

$5 Signup and view all the answers

Half of all potential customers would pay $10 for your product but half would pay $8. Your marginal costs are $4. If you set the price at $10, the expected profit is:

$3 Signup and view all the answers

A movie executive must decide to fund a new movie project. A success would earn $20 million and a failure would cost $60 million in lost profits. At what probability of expected success should he fund the movie?

0.75 Signup and view all the answers

A pharmaceutical company must decide whether to fund a new drug development project. A success would earn $10 million and a failure would cost $90 million. At what probability of expected success should she fund the project?

0.10 Signup and view all the answers

Which types of poor decisions are more visible to a decision maker's supervisor?

Agreeing to undertake a project Signup and view all the answers

The key distinction between risk and uncertainty is:

Uncertainty refers to not knowing possible outcomes or their probabilities Signup and view all the answers

The expected value of a gamble that pays $10 with probability 0.2 and -$4 with probability 0.8 is:

$-1.20 Signup and view all the answers

You want to run a difference-in-difference experiment with a price increase for your lawn chairs in Miami. If you are worried about the 'representativeness' of your control group, a good comparison city would be:

Tampa Bay Signup and view all the answers

You want to run a difference-in-difference experiment with a price increase for the bacon cheeseburger item on your menu. If you are worried about 'leakage' with your control group, a good comparison menu item would be:

Soft drink sales Signup and view all the answers

A false positive is:

When you incorrectly conclude that your hypothesis is true Signup and view all the answers

A false negative is:

When you incorrectly conclude that your hypothesis is false Signup and view all the answers

We worry that false negatives occur too often relative to false positives due to:

Managers having an incentive to make a false negative conclusion because they are harder for superiors to observe Signup and view all the answers

What are the Type I and Type II error costs?

$490,000 and $150,000 Signup and view all the answers

Based on expected error costs, you should fund a project if it has better than what percentage chance of success?

60% Signup and view all the answers

What should you do if the average probability from studies is less than your threshold for funding?

You should not open the store. Signup and view all the answers

When devising a strategy with considerable uncertainty, what is the recommended approach?

Leave room for improvisation. Signup and view all the answers

What is the expected value of an uncertain outcome calculated as?

The sum of probabilities times the random variables Signup and view all the answers

What should you consider if your uncle offers you a deal with an expected value much greater than the cost?

All of the above. Signup and view all the answers

If you calculate the expected value of option A is greater than option B by 0.013%, what is your likely course of action?

Flip a coin Signup and view all the answers

When deciding whether to sell widgets for $5 or $6, which price yields higher profits?

Both prices earn the same profits Signup and view all the answers

You might increase your profits by:

Setting a $6 price but offering a $1 discount Signup and view all the answers

Selection bias occurs because:

There are systematic differences between treatment and control groups Signup and view all the answers

Why are randomized experiments considered the 'gold standard'?

Selection bias is eliminated Signup and view all the answers

What is the key determinant of a careful observational study?

The control group mimics the treatment group Signup and view all the answers

If you reject a correct hypothesis, what error are you committing?

Type I error Signup and view all the answers

A manager will decide to fund too few projects if:

All of the above. Signup and view all the answers

What is the difference between risk and uncertainty?

None of the above. Signup and view all the answers

How can you deal with uncertainty effectively?

Add flexibility to allow responses Signup and view all the answers

What is the expected value of guessing on a multiple-choice test with five options?

0 points Signup and view all the answers

What is the expected value of NPV when considering a new store with different probabilities?

$4.0 million Signup and view all the answers

What is the expected profit or loss from adding a new manufacturing line with varying probabilities?

Gain of $47,000 Signup and view all the answers

What is the expected NPV of a viral mobile app investment?

$0 Signup and view all the answers

What is the minimum probability of success required to invest in a project with an NPV of $75 million?

1/4 Signup and view all the answers

What do you estimate the referrals per dollar are through agency A?

1.1 referrals per dollar Signup and view all the answers

What is the probability that a random customer will use the coupon if 90 out of 3,000 responded?

0.03 Signup and view all the answers

What are the expected error costs of shutting down process B first?

$16,000 Signup and view all the answers

What price maximizes expected revenue if 40% value it at $10 and 60% at $6?

$6 Signup and view all the answers

You should enter the market if you believe the probability that the incumbent will price low is less than what percentage?

0.75 Signup and view all the answers

Study Notes

Random Variables and Expected Value

A random variable can take on different values.
The expected value is calculated as the sum of the products of probabilities and outcomes.

Expected Value in Gamble Scenarios

If winning $1 on heads and losing $1 on tails, expected value is $0.
Winning $2 on heads and losing $1 on tails results in an expected value of $0.50.
Winning $3 on heads and losing $1 on tails leads to an expected value of $1.00.

Investment Decisions

Comparing projects A and B indicates investing in project A is favorable when A's expected success probability is 0.7.
If project A has a 0.5 probability of success, it's indeterminate if A or B should be chosen.

Calculating Expected Values

For three equally likely outcomes with payoffs of $3, $6, and $9, the expected value is $6.
The expected value of four payoffs of $2, $4, $6, and $8 is $5.

Profit Calculations

If high-value customers pay $200 and low-value customers pay $175, expected profit per customer is $75 when marginal costs are $100.

Experiment Methodologies

Difference-in-difference estimates compare changes in outcomes between treatment and control groups to assess effects like price changes or advertising increases.

Cost-Benefit Analyses

For decisions such as funding projects with expected success, funding is justified when the probability exceeds specific thresholds (e.g., 0.75 for a movie project).

Error Types in Hypotheses

A "false positive" occurs when a hypothesis is incorrectly accepted.
Conversely, a "false negative" is the incorrect rejection of a hypothesis.

Risk and Uncertainty

Risk is quantifiable, while uncertainty involves unknown potential outcomes and their probabilities.

Managerial Decisions

An inclination to avoid false positives often influences managerial decision-making patterns.
Decisions made by managers may be influenced by the perceived visibility of outcomes, with major decisions likely scrutinized more heavily.

Strategies for Uncertainty

In uncertain situations, develop flexible strategies that allow for improvisation rather than rigid plans that fail to adapt to unforeseen events.

Calculation Framework

The expected value is derived from summing the product of probabilities and their associated values to inform decision-making.### Project Valuation Considerations
Valuation may underestimate project value in worst-case scenarios.
Success likelihood of the deal may be overvalued.

Decision-Making in Expected Values

If the expected value of option A is only slightly higher than option B, it’s reasonable to flip a coin instead of choosing definitively.

Pricing Strategies for Widgets

Selling at $5 and $6 with a marginal cost of $3 leads to equal profits due to different success rates.
Higher profits may be achieved by strategically pricing at $6 with a discount when negotiations stall.

Understanding Selection Bias

Selection bias arises from systematic differences between treatment and control groups impacting outcomes.

Importance of Randomized Controlled Experiments

These experiments eliminate selection bias, ensuring no systematic differences affect treatment and control groups.

Observational Study Quality

A strong observational study must ensure the control group mirrors the treatment group without systematic differences.

Hypothesis Testing Errors

Rejecting a true hypothesis results in a Type I error, indicating a false positive.

Project Funding Decisions

Managers may underfund uncertain projects due to known failures, lack of exposure to passed opportunities, and inadequate review processes.

Risk vs. Uncertainty

Neither risk nor uncertainty is accurately explained by typical definitions; they are distinct concepts in decision analysis.

Flexibility in Uncertain Situations

Strategies should incorporate flexibility to adapt to unforeseen contingencies.

Expected Value in Random Guessing

Random guessing on a multiple-choice test results in an expected value of zero, considering correct and incorrect answers.

Valuation of New Store Opportunities

The expected NPV for a new store considering varying population growth scenarios is estimated at $4 million.

Expected Profit/Loss Calculation

A new manufacturing line could yield an expected gain of $47,000 based on calculated probabilities.

NPV of a New Mobile App Investment

Expected NPV for a mobile app, factoring in probability of success and failure scenarios, results in an NPV of zero.

Investment Decision Threshold

For a project with potential NPVs of $75 million and -$25 million, a minimum success probability of 25% is necessary for investment justification.

Referral Effectiveness in Advertising

Agency A provides an estimated 1.1 referrals per dollar spent, based on increased ad spend and changes in referrals.

Probability of Coupon Usage

In a customer list of 3,000, the probability of coupon usage is 3%, based on past response rates.

Error Costs in Process Investigation

Investigating process A first has expected error costs of $16,000 versus lower costs for process B.

Pricing for Maximum Revenue

The optimal price to maximize expected revenue is $6, catering to varying buyer valuations.

Market Entry Considerations

Entering a market is justifiable if there is a belief that the incumbent’s likelihood of pricing low is below 75%.

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Description

This quiz covers the concepts of random variables and expected value, including their applications in gambling and investment decisions. It explores how to calculate expected values in various scenarios and provides examples to illustrate these principles.