Solved Exercises on Expected Value in Algebra 2

Master expected value: probability distributions, weighted averages, decision-making through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Basic Expected Value
Exercise 1
A game involves rolling a fair six-sided die. If you roll a 1, you win $5. If you roll a 2 or 3, you win $2. If you roll a 4, 5, or 6, you lose $3. Find the expected value of one play.
Definition:

Expected Value: \(E(X) = \sum_{i} x_i \cdot P(x_i)\) where \(x_i\) are outcomes and \(P(x_i)\) are their probabilities

Expected value calculation method:
  1. List all possible outcomes and their associated values
  2. Determine the probability of each outcome
  3. Multiply each outcome by its probability
  4. Sum all the products to get the expected value
Outcomes
Roll 1: +$5, Roll 2-3: +$2, Roll 4-6: -$3
Probabilities
P(1)=1/6, P(2-3)=2/6, P(4-6)=3/6
Expected Value
$0.50
Step 1: Identify all possible outcomes and values

Roll 1: Win $5, Roll 2 or 3: Win $2, Roll 4, 5, or 6: Lose $3

Step 2: Determine probabilities

P(Roll 1) = 1/6, P(Roll 2 or 3) = 2/6, P(Roll 4, 5, or 6) = 3/6

Step 3: Calculate expected value

E(X) = (5 × 1/6) + (2 × 2/6) + (-3 × 3/6)

E(X) = 5/6 + 4/6 - 9/6 = 0/6 = $0

E(X) = $0
Final answer:

The expected value is $0, meaning the game is fair in the long run.

Applied rules:

• Expected Value Formula: Sum of outcomes multiplied by their probabilities

• Weighted Average: Each outcome weighted by its likelihood

• Long-term Average: What you'd expect per trial over many trials

2 Probability Distribution
Exercise 2
A probability distribution is given as follows:
X = {0, 1, 2, 3} with probabilities P(X) = {0.2, 0.3, 0.4, 0.1}.
Find the expected value.
Definition:

Probability Distribution: A table showing all possible values of a random variable and their corresponding probabilities

Distribution
X={0,1,2,3}, P(X)={0.2,0.3,0.4,0.1}
Calculation
E(X) = 0×0.2 + 1×0.3 + 2×0.4 + 3×0.1
Result
E(X) = 1.4
Step 1: Set up the probability distribution table
X 0 1 2 3
P(X) 0.2 0.3 0.4 0.1
Step 2: Apply the expected value formula

E(X) = Σ[x_i × P(x_i)] = (0 × 0.2) + (1 × 0.3) + (2 × 0.4) + (3 × 0.1)

Step 3: Calculate each product

(0 × 0.2) = 0, (1 × 0.3) = 0.3, (2 × 0.4) = 0.8, (3 × 0.1) = 0.3

Step 4: Sum all products

E(X) = 0 + 0.3 + 0.8 + 0.3 = 1.4

E(X) = 1.4
Final answer:

The expected value is 1.4, which represents the average value over many trials.

Applied rules:

• Probability Distribution: All probabilities must sum to 1

• Expected Value: Weighted average of all possible outcomes

• Linear Property: E(aX + b) = aE(X) + b

3 Real-world Application
Exercise 3
A company estimates that a project has a 30% chance of earning $100,000 profit, a 50% chance of earning $50,000 profit, and a 20% chance of losing $25,000. What is the expected profit?
Definition:

Expected Profit: The average profit over many repetitions of the same scenario

Scenarios
30%→+$100K, 50%→+$50K, 20%→-$25K
Formula
E(P) = Σ[profit × probability]
Result
$50,000
Step 1: Identify all possible outcomes

Outcome 1: $100,000 profit with probability 0.3

Outcome 2: $50,000 profit with probability 0.5

Outcome 3: -$25,000 loss with probability 0.2

Step 2: Apply expected value formula

E(Profit) = ($100,000 × 0.3) + ($50,000 × 0.5) + (-$25,000 × 0.2)

Step 3: Calculate each product

($100,000 × 0.3) = $30,000

($50,000 × 0.5) = $25,000

(-$25,000 × 0.2) = -$5,000

Step 4: Sum all products

E(Profit) = $30,000 + $25,000 - $5,000 = $50,000

E(Profit) = $50,000
Final answer:

The expected profit is $50,000, which represents the average outcome if the project were repeated many times.

Applied rules:

• Decision Making: Positive expected value suggests favorable outcome

• Risk Assessment: Expected value helps evaluate potential investments

• Probability Weighting: More likely outcomes contribute more to expected value

Rules and methods, laws,...
\(E(X) = \sum_{i} x_i \cdot P(x_i)\)
Expected Value Formula
Linearity
\(E(aX + b) = aE(X) + b\)
Expected value of linear transformation
Additivity
\(E(X + Y) = E(X) + E(Y)\)
Expected value of sums
Constant
\(E(c) = c\)
Expected value of constant
Key definitions:

Random Variable: A variable whose possible values are outcomes of a random phenomenon

Probability Distribution: A function showing probabilities of all possible outcomes

Expected Value: The weighted average of all possible outcomes, where weights are probabilities

Complete methodology:
  1. Identify Outcomes: List all possible values of the random variable
  2. Determine Probabilities: Assign probability to each outcome
  3. Calculate Products: Multiply each outcome by its probability
  4. Sum Results: Add all products to get expected value
Tip 1: Always verify that probabilities sum to 1 before calculating expected value.
Tip 2: Negative outcomes represent losses or costs in financial applications.
Tip 3: Expected value may not be one of the actual possible outcomes.
Tip 4: Use expected value for decision-making in uncertain scenarios.
Common errors: Forgetting to multiply by probabilities, miscalculating probabilities, ignoring negative outcomes.
Exam preparation: Practice with various probability distributions, understand real-world applications, memorize formulas.
Formulas to know by heart:

• Expected Value: \(E(X) = \sum_{i} x_i \cdot P(x_i)\)

• Linear Transformation: \(E(aX + b) = aE(X) + b\)

• Additivity: \(E(X + Y) = E(X) + E(Y)\)

• Constant: \(E(c) = c\)

Solution: Exercises 4 to 5
4 Lottery Problem
Exercise 4
A lottery ticket costs $2. There's a 1 in 1000 chance to win $500, a 1 in 100 chance to win $10, and a 989 in 1000 chance of winning nothing. What is the expected value of buying one ticket?
Definition:

Net Gain/Loss: Calculate expected value considering the cost of participation

Cost
$2 ticket
Outcomes
Win $500, Win $10, Win $0
Expected Value
-$1.40
Step 1: Convert probabilities to decimals

P(Win $500) = 1/1000 = 0.001

P(Win $10) = 1/100 = 0.01

P(Win $0) = 989/1000 = 0.989

Step 2: Calculate net gains for each outcome

Net gain if win $500: $500 - $2 = $498

Net gain if win $10: $10 - $2 = $8

Net gain if win $0: $0 - $2 = -$2

Step 3: Apply expected value formula

E(X) = ($498 × 0.001) + ($8 × 0.01) + (-$2 × 0.989)

Step 4: Calculate each product

($498 × 0.001) = $0.498

($8 × 0.01) = $0.08

(-$2 × 0.989) = -$1.978

Step 5: Sum all products

E(X) = $0.498 + $0.08 - $1.978 = -$1.40

E(X) = -$1.40
Final answer:

The expected value is -$1.40, meaning you lose an average of $1.40 per ticket bought.

Applied rules:

• Net Gain: Subtract cost from winnings to get true value

• Expected Loss: Negative expected value indicates long-term loss

• Lottery Analysis: Most lotteries have negative expected value

5 Investment Decision
Exercise 5
An investment has a 25% chance of returning 15% profit, a 60% chance of returning 5% profit, and a 15% chance of losing 10%. If you invest $1000, what is the expected value of your investment after one year?
Definition:

Investment Return: Calculate final amount including original investment

Initial Investment
$1000
Returns
15%, 5%, -10%
Expected Value
$1040
Step 1: Calculate final amounts for each scenario

Scenario 1 (15% profit): $1000 × 1.15 = $1150

Scenario 2 (5% profit): $1000 × 1.05 = $1050

Scenario 3 (10% loss): $1000 × 0.90 = $900

Step 2: Apply expected value formula

E(Final Amount) = ($1150 × 0.25) + ($1050 × 0.60) + ($900 × 0.15)

Step 3: Calculate each product

($1150 × 0.25) = $287.50

($1050 × 0.60) = $630.00

($900 × 0.15) = $135.00

Step 4: Sum all products

E(Final Amount) = $287.50 + $630.00 + $135.00 = $1052.50

Step 5: Calculate expected profit

Expected Profit = $1052.50 - $1000 = $52.50

E(Final Amount) = $1052.50
Final answer:

The expected value of the investment after one year is $1052.50, representing an expected profit of $52.50.

Applied rules:

• Investment Analysis: Expected value helps compare different investment options

• Return Calculation: Include both principal and returns in final amount

• Decision Making: Positive expected value indicates potentially favorable investment

Expected Value Fundamentals & Applications
\(E(X) = \sum_{i} x_i \cdot P(x_i)\)
Expected Value Formula
Key definitions:

Expected Value: The long-run average value of repetitions of an experiment

Random Variable: A variable whose value depends on outcomes of a random phenomenon

Discrete Distribution: A distribution with countable number of possible outcomes

Complete methodology:
  1. Identify Outcomes: List all possible values of the random variable
  2. Determine Probabilities: Assign probability to each outcome
  3. Calculate Products: Multiply each outcome by its probability
  4. Sum Results: Add all products to get expected value
  5. Interpret Results: Understand practical meaning of the expected value
Tip 1: Always check that probabilities sum to 1 before calculating expected value.
Tip 2: Expected value can be a decimal even when outcomes are whole numbers.
Tip 3: In games, positive expected value favors the player; negative favors the house.
Tip 4: Expected value represents long-term average, not guaranteed outcome for single trial.
Applications: Insurance premiums, stock market analysis, game theory, quality control, risk assessment.
Properties: E(aX + b) = aE(X) + b, E(X + Y) = E(X) + E(Y), E(c) = c.
Essential formulas:

• Expected Value: \(E(X) = \sum_{i} x_i \cdot P(x_i)\)

• Linear Transformation: \(E(aX + b) = aE(X) + b\)

• Sum of Variables: \(E(X + Y) = E(X) + E(Y)\)

• Constant: \(E(c) = c\)

Expected Value Visualization
Exercise 6: Comparing Expected Values
Compare the expected values of two investment options:
Option A: 50% chance of +$100, 50% chance of -$50
Option B: 25% chance of +$200, 75% chance of -$25

Analysis: The chart compares the expected values of different investment strategies.

  • Option A: E(X) = (100 × 0.5) + (-50 × 0.5) = $25
  • Option B: E(X) = (200 × 0.25) + (-25 × 0.75) = $31.25

Questions & Answers

Question: I don't understand why the expected value might not be one of the actual possible outcomes. Can you explain?

Answer: Great question! The expected value is a weighted average, not necessarily a possible outcome. Think of it this way:

  • When you flip a coin twice, possible outcomes are 0, 1, or 2 heads
  • But the expected number of heads is 1 (which is possible)
  • However, if you consider rolling a die, possible outcomes are 1, 2, 3, 4, 5, 6
  • The expected value is 3.5 (which is impossible to actually roll)

The expected value represents the long-term average you'd expect if you repeated the experiment many times. It's a theoretical average, not a guaranteed result for any single trial.

In our dice example, if you rolled a die thousands of times, your average result would approach 3.5, even though you can never actually roll a 3.5.

Question: How do I interpret negative expected values in business decisions?

Answer: Negative expected values in business contexts indicate that on average, you'll lose money over many repetitions of the same scenario:

  • If an investment has an expected value of -$500, you'd lose $500 on average per investment over many trials
  • This doesn't mean you'll lose exactly $500 each time, but rather that losses will average $500 per investment
  • Many insurance policies have negative expected values for customers (positive for the company)
  • Lottery tickets typically have strongly negative expected values

However, businesses might still engage in activities with slightly negative expected values if they provide other benefits like customer loyalty, brand exposure, or portfolio diversification.

The key is understanding that negative expected value means a long-term loss, but individual outcomes can still be profitable.

Question: What's the relationship between expected value and variance? Are they equally important?

Answer: Expected value and variance measure different aspects of a distribution and are both important:

  • Expected Value: Measures the central tendency or average outcome
  • Variance: Measures the spread or variability of outcomes around the mean
  • Two investments could have the same expected value but very different variances
  • Higher variance means greater risk and uncertainty

For example, Investment A might have E(X) = $1000 with low variance (consistent returns), while Investment B has E(X) = $1000 with high variance (unpredictable returns).

Most investors prefer higher expected value with lower variance, but risk tolerance varies. The expected value tells you what to expect on average, while variance tells you how much uncertainty there is around that expectation.

In decision making, both metrics are crucial: expected value for potential reward, variance for risk assessment.

Question: How do I handle problems where outcomes are percentages rather than absolute values?

Answer: When dealing with percentage returns, convert them to actual dollar amounts based on your initial investment:

For example, if you invest $1000 and have a 30% chance of gaining 10% and a 70% chance of losing 5%:

  • 10% gain = $1000 × 0.10 = $100 gain
  • -5% loss = $1000 × -0.05 = $50 loss
  • Then apply expected value: E(X) = ($100 × 0.3) + (-$50 × 0.7) = $30 - $35 = -$5

Alternatively, you can work with percentage returns directly and then apply to your investment:

  • E(Return %) = (10% × 0.3) + (-5% × 0.7) = 3% - 3.5% = -0.5%
  • Expected dollar value = $1000 × (1 + (-0.005)) = $995

Both methods yield the same result. Choose the approach that feels more intuitive for the specific problem.