SAT Math Lesson Plan 14: Advanced Probability and Expected Value

Lesson Summary

Lesson 14 in this SAT Math Prep series focuses on advanced probability concepts and expected value—topics that make up roughly 5–7% of the SAT Math section. Students will learn how to interpret probability distributions, compute expected values using weighted averages, and apply these ideas in real-world decision-making contexts such as finance, insurance, and games of chance. Through examples, practice problems, and a quiz, students build fluency in applying mathematical reasoning to uncertain outcomes. This 45-minute lesson includes a Warm-Up, six worked examples, a Review, and a 10-question multiple-choice quiz with answer key. Aligned with Common Core standards, this lesson supports SAT success while reinforcing key statistical reasoning skills.

Lesson Objectives

• Define and apply probability distributions.
• Calculate expected value and interpret its significance.
• Use probability models in financial and strategic decision-making.
• Evaluate games and scenarios using fairness and expected outcomes.

Common Core Standards

• CCSS.MATH.CONTENT.HSS.MD.A.1 – Define a random variable for a quantity of interest and graph its probability distribution.
• CCSS.MATH.CONTENT.HSS.MD.A.2 – Calculate expected values and interpret them as average outcomes.
• CCSS.MATH.CONTENT.HSS.MD.B.5 – Use expected values to solve problems in decision-making.

Prerequisite Skills

• Basic understanding of probability rules.
• Comfort working with fractions, decimals, and percentages.
• Familiarity with probability distributions and weighted averages.

Key Vocabulary

• Probability Distribution: A function that shows the possible values of a random variable and their probabilities.
• Expected Value (EV): The long-term average outcome of a probability experiment.
• Random Variable: A variable whose values depend on the outcome of a random event.
• Weighted Average: A mean where some values contribute more than others based on their probabilities.
• Fair Game: A game in which the expected value is zero, meaning no player has an advantage.
• Risk Assessment: The process of evaluating potential outcomes using probability models.

Warm Up

This activity reviews basic probability and sets the stage for understanding expected value.

Problem: A game involves rolling a fair six-sided die. If the die shows a 6, you win \$5. Otherwise, you win nothing. What is the expected payout?

Step 1:

Determine all possible outcomes and their probabilities:
- Probability of rolling a 6: \( \frac{1}{6} \)
- Probability of not rolling a 6: \( \frac{5}{6} \)

Step 2:

Assign payouts to each outcome:
- Win \$5 with probability \( \frac{1}{6} \)
- Win \$0 with probability \( \frac{5}{6} \)

Step 3:

Calculate the expected value: \[ EV = (5 \times \frac{1}{6}) + (0 \times \frac{5}{6}) = \frac{5}{6} \approx 0.83 \]

Answer: The expected payout is approximately \$0.83.

Self-Study Tip: If you were to play this game many times, you'd win \$5 about 1 out of every 6 games. Expected value represents that long-term average.

Teach

Expected value questions require you to combine your understanding of probability with weighted averages. The SAT may ask you to compute the expected outcome of a financial decision, a game scenario, or a probability distribution. You might also be asked to assess whether a situation is “fair” or which option has the better long-term return.

In this section, you’ll explore different types of expected value problems, including:

• Discrete probability models
• Multi-outcome games
• Financial decisions involving risk
• Comparing scenarios for fairness and strategy

Each example will walk you through the process of calculating expected value, interpreting the result, and applying the concept to real-world reasoning.

Example 1: Basic Expected Value from a Probability Distribution

This example introduces how to compute expected value using a basic distribution.

Problem: A random variable \( X \) represents the outcome of a spinner with the following payouts:

Step 1:

Multiply each outcome by its payout:
\[ EV = (4 \times \frac{1}{4}) + (2 \times \frac{1}{2}) + (0 \times \frac{1}{4}) = 1 + 1 + 0 = 2 \]

Answer: The expected value is \$2.

Example 2: Game with Unequal Payouts

This example shows how unequal winnings affect the expected outcome.

Problem: You pay \$3 to play a game. A bag contains 5 red marbles and 1 blue marble. If you draw the blue marble, you win \$20. Otherwise, you win nothing. What is the expected value of this game?

Step 1:

Find the probabilities:
- \( P(\text{blue}) = \frac{1}{6} \)
- \( P(\text{red}) = \frac{5}{6} \)

Step 2:

Calculate the expected value of the winnings: \[ EV = (20 \times \frac{1}{6}) + (0 \times \frac{5}{6}) = \frac{20}{6} \approx 3.33 \]

Step 3:

Subtract the cost to play: \[ \text{Net EV} = 3.33 - 3 = 0.33 \]

Answer: The expected value of playing the game is approximately \$0.33.

Example 3: Interpreting Expected Value in Context

This example emphasizes how expected value informs real-world decisions.

Problem: A company offers a promotional scratch-off card. You have a 1 in 50 chance of winning \$100, and otherwise you win nothing. Should you be willing to pay \$3 for this promotion?

Step 1:

\[ EV = (100 \times \frac{1}{50}) + (0 \times \frac{49}{50}) = 2 + 0 = 2 \]

Step 2:

Compare the expected winnings to the cost:
\[ \text{Net EV} = 2 - 3 = -1 \]

Answer: The expected value is negative, so on average, you lose \$1. This is not a fair deal from the player's perspective.

Self-Study Tip: A negative expected value means that over many plays, you’d lose money. Always compare EV to cost.

Example 4: Multi-Outcome Game

This example introduces expected value with more than two outcomes.

Problem: In a game, you roll a fair six-sided die. If you roll a 6, you win \$10. If you roll a 5, you win \$5. Any other number wins nothing. What is the expected value of this game?

Step 1:

List outcomes and probabilities:
- P(6) = \( \frac{1}{6} \), payout = \$10
- P(5) = \( \frac{1}{6} \), payout = \$5
- P(1–4) = \( \frac{4}{6} \), payout = \$0

Step 2:

Calculate expected value: \[ EV = (10 \times \frac{1}{6}) + (5 \times \frac{1}{6}) + (0 \times \frac{4}{6}) = \frac{10 + 5}{6} = \frac{15}{6} = 2.5 \]

Answer: The expected value of the game is \$2.50.

Example 5: Strategic Decision Using Expected Value

This example shows how expected value can guide a decision between two options.

Problem: You have two options:

1. Take a guaranteed \$6
2. Flip a coin: heads you win \$15, tails you win \$0

Which is the better option based on expected value?

Step 1:

Calculate EV for Option B: \[ EV = (15 \times \frac{1}{2}) + (0 \times \frac{1}{2}) = 7.5 \]

Step 2:

Compare to Option A’s guaranteed \$6

Answer: Option B has a higher expected value (\$7.50), even though it's riskier. Based on EV alone, Option B is the better choice.

Self-Study Tip: On the SAT, expected value is a mathematical average, not necessarily a recommendation—always consider context and risk.

Example 6: Is the Game Fair?

This example introduces the idea of a fair game, where EV = 0.

Problem: A carnival game costs \$2 to play. You spin a wheel and win \$10 with probability \( \frac{1}{5} \). You win nothing otherwise. Is this a fair game?

Step 1:

\[ EV = (10 \times \frac{1}{5}) + (0 \times \frac{4}{5}) = 2 \]

Step 2:

\[ \text{Net EV} = 2 - 2 = 0 \]

Answer: Yes, this is a fair game because the expected value equals the cost to play. On average, you neither gain nor lose money.

Review

In this lesson, you learned how to compute and interpret expected value across a variety of contexts, including discrete distributions, games of chance, and strategic decision-making scenarios. You also learned to evaluate fairness in probabilistic settings and to compare outcomes using mathematical reasoning. These types of questions appear occasionally on the SAT, particularly in multi-step word problems involving probability and algebraic thinking.

Here’s a summary of what was covered:

• How to define and use a probability distribution
• How to calculate expected value using outcomes and probabilities
• How to assess the fairness of a game or decision based on expected return
• How to make informed decisions by comparing EVs

Example 1: Comparing Two Scenarios

Problem: Two games offer the following outcomes:

• Game A: 1 in 10 chance to win \$20
• Game B: 1 in 5 chance to win \$9

Which game has the higher expected value?

Step 1:

Game A EV = \( \frac{1}{10} \times 20 = 2 \)
Game B EV = \( \frac{1}{5} \times 9 = 1.8 \)

Answer: Game A has the higher expected value.

Example 2: Cost vs. Expected Value

Problem: You pay \$3 to play a game where you roll a die. If you roll a 6, you win \$12. Otherwise, you win nothing. Is this a good deal?

Step 1:

EV = \( \frac{1}{6} \times 12 = 2 \)
Net EV = 2 - 3 = -1

Answer: This is not a good deal. On average, you lose \$1 each time you play.

Example 3: Expected Value from a Distribution Table

Problem: A variable \( X \) has the following distribution:

Step 1:

\[ EV = (1 \times \frac{1}{5}) + (2 \times \frac{2}{5}) + (4 \times \frac{2}{5}) = \frac{1}{5} + \frac{4}{5} + \frac{8}{5} = \frac{13}{5} = 2.6 \]

Answer: The expected value of \( X \) is 2.6.

Example 3: Expected Value from a Distribution Table

Problem: A variable \( X \) has the following distribution:

Step 1:

\[ EV = (1 \times \frac{1}{5}) + (2 \times \frac{2}{5}) + (4 \times \frac{2}{5}) = \frac{1}{5} + \frac{4}{5} + \frac{8}{5} = \frac{13}{5} = 2.6 \]

Answer: The expected value of \( X \) is 2.6.

Multimedia Resources

For additional support with advanced probability and expected value—including interactive tutorials, videos, and practice problems—explore this collection of aligned resources from Media4Math:

https://www.media4math.com/LessonPlans/SupportResourcesSATMathLesson14

Quiz

Directions: Choose the best answer for each question. Show your work on a separate sheet if needed.

1. A game pays \$10 if you roll a 6 on a fair die. Otherwise, you win nothing. What is the expected value?
   a) \$1.67
   b) \$1.00
   c) \$2.00
   d) \$3.33
2. You flip a coin. If heads, you win \$4. If tails, you lose \$2. What is the expected value?
   a) \$1.00
   b) \$0.50
   c) \$2.00
   d) \$0.00
3. A spinner has 4 equal sections labeled \$0, \$1, \$2, and \$3. What is the expected value of one spin?
   a) \$1.00
   b) \$1.50
   c) \$2.00
   d) \$3.00
4. A raffle ticket costs \$5. There is a 1 in 50 chance to win \$100. What is the expected value of playing?
   a) \$2.00
   b) \$1.00
   c) -\$3.00
   d) -\$4.00
5. Which of the following is a fair game?
   a) Pay \$2 to win \$5 with probability \( \frac{1}{5} \)
   b) Pay \$3 to win \$6 with probability \( \frac{1}{3} \)
   c) Pay \$1 to win \$4 with probability \( \frac{1}{4} \)
   d) Pay \$2 to win \$10 with probability \( \frac{1}{10} \)
6. A company offers a bonus: \$500 with probability 0.2, \$0 otherwise. What is the expected bonus?
   a) \$100
   b) \$200
   c) \$250
   d) \$400
7. A quiz offers 3 points for a correct answer (70% chance), 0 for an incorrect answer. What is the expected value per question?
   a) 2.1
   b) 1.5
   c) 1.0
   d) 0.9
8. You pay \$2 to play a game that returns \$5 with probability \( \frac{1}{4} \). What is the expected value?
   a) \$1.25
   b) \$1.00
   c) \$0.25
   d) -\$0.75
9. Which of the following would result in a negative expected value?
   a) EV = 3, cost = 4
   b) EV = 4, cost = 4
   c) EV = 5, cost = 4
   d) EV = 6, cost = 4
10. A variable \( X \) takes values 2, 3, and 5 with probabilities \( \frac{1}{5} \), \( \frac{2}{5} \), and \( \frac{2}{5} \) respectively. What is the expected value of \( X \)?
    a) 3.0
    b) 3.4
    c) 3.6
    d) 4.0

Answer Key

1. Answer: a) \$1.67
   \[ EV = \frac{1}{6} \times 10 = \frac{10}{6} \approx 1.67 \]
2. Answer: b) \$0.50
   \[ EV = \frac{1}{2} \times 4 + \frac{1}{2} \times (-2) = 2 - 1 = 1 \]
3. Answer: b) \$1.50
   \[ EV = \frac{1}{4}(0 + 1 + 2 + 3) = \frac{6}{4} = 1.5 \]
4. Answer: c) -\$3.00
   \[ EV = \frac{1}{50} \times 100 = 2 \quad \text{Net EV} = 2 - 5 = -3 \]

5. Answer: b) Pay \$3 to win \$6 with probability \( \frac{1}{3} \)
   \[ EV = \frac{1}{3} \times 6 = 2 \quad \text{Net EV} = 2 - 3 = -1 \quad \text{(Wait! Wrong!)}
   Let's check all options:

   • a) EV = 1 → -1 loss
   • b) EV = 2 → -1 loss
   • c) EV = 1 → 0
   • d) EV = 1 → -1

   So the correct answer is c): EV = \$1, cost = \$1 → fair game.

6. Answer: a) \$100
   \[ EV = 0.2 \times 500 = 100 \]
7. Answer: a) 2.1
   \[ EV = 0.7 \times 3 + 0.3 \times 0 = 2.1 + 0 = 2.1 \]
8. Answer: d) -\$0.75
   \[ EV = \frac{1}{4} \times 5 = 1.25 \quad \text{Net EV} = 1.25 - 2 = -0.75 \]
9. Answer: a) EV = 3, cost = 4 → -1
10. Answer: c) 3.6
    \[ EV = 2 \times \frac{1}{5} + 3 \times \frac{2}{5} + 5 \times \frac{2}{5} = \frac{2}{5} + \frac{6}{5} + \frac{10}{5} = \frac{18}{5} = 3.6 \]