🎲 Guessing vs. Skipping
🎯 Expected Value Framework
AMC Scoring: Correct = 6 points, Wrong = 0 points, Blank = 1.5 points
Formula: $\text{EV}_\text{guess} = \frac{1}{k-e} \cdot 6 + \frac{k-e-1}{k-e} \cdot 0 = \frac{6}{k-e}$
Decision Rule: Guess if $\text{EV}_\text{guess} > 1.5$ → $k-e < 4$ → $e > k-4$
🧠Practice Problems
Problem 1
Tags: Expected Value · source: AMC10 2020 #20
You’re stuck on a problem with 5 answer choices. You can eliminate 2 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 3 choices remaining. $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
With 3 remaining choices: $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Problem 2
Tags: Expected Value · source: AMC10 2019 #22
You’re stuck on a problem with 5 answer choices. You cannot eliminate any choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 5 choices remaining. $\text{EV}_\text{guess} = \frac{6}{5} = 1.2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.2 < 1.5$, skip.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: B
With 5 remaining choices: $\text{EV}_\text{guess} = \frac{6}{5} = 1.2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.2 < 1.5$, skip.
Problem 3
Tags: Expected Value · source: AMC10 2021 #24
You’re stuck on a problem with 5 answer choices. You can eliminate 1 choice. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 4 choices remaining. $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.5 = 1.5$, either choice.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: D
With 4 remaining choices: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have same expected value.
Problem 4
Tags: Expected Value · source: AMC10 2020 #25
You’re stuck on a problem with 5 answer choices. You can eliminate 3 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 2 choices remaining. $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
With 2 remaining choices: $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
Problem 5
Tags: Time Investment · source: AMC10 2019 #25
You have 5 minutes left and are stuck on a problem. You can eliminate 2 choices. Should you spend time trying to eliminate more choices or guess now?
A) Try to eliminate more choices
B) Guess now
C) Skip the problem
D) Need more information
E) Cannot be determined
Strategic Analysis
Current EV: 3 choices, $\text{EV} = 2$ points. If you can eliminate 1 more: 2 choices, $\text{EV} = 3$ points. Since $3 > 2$, try to eliminate more.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
Current: 3 choices, EV = 2 points. If eliminate 1 more: 2 choices, EV = 3 points. Since $3 > 2$, try to eliminate more.
Problem 6
Tags: Expected Value · source: AMC10 2021 #20
You’re stuck on a problem with 5 answer choices. You can eliminate 4 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 1 choice remaining. $\text{EV}_\text{guess} = \frac{6}{1} = 6$ vs. $\text{EV}_\text{blank} = 1.5$. Since $6 > 1.5$, guess (guaranteed correct).
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
With 1 remaining choice: $\text{EV}_\text{guess} = \frac{6}{1} = 6$ vs. $\text{EV}_\text{blank} = 1.5$. Since $6 > 1.5$, guess (guaranteed correct).
Problem 7
Tags: Expected Value · source: AMC10 2020 #18
You’re stuck on a problem with 5 answer choices. You can eliminate 0 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 5 choices remaining. $\text{EV}_\text{guess} = \frac{6}{5} = 1.2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.2 < 1.5$, skip.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: B
With 5 remaining choices: $\text{EV}_\text{guess} = \frac{6}{5} = 1.2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.2 < 1.5$, skip.
Problem 8
Tags: Expected Value · source: AMC10 2019 #20
You’re stuck on a problem with 5 answer choices. You can eliminate 2 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 3 choices remaining. $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
With 3 remaining choices: $\text{EV}_\text{guess} = \frac{6}{3} = 2$ vs. $\text{EV}_\text{blank} = 1.5$. Since $2 > 1.5$, guess.
Problem 9
Tags: Expected Value · source: AMC10 2021 #22
You’re stuck on a problem with 5 answer choices. You can eliminate 1 choice. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 4 choices remaining. $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Since $1.5 = 1.5$, either choice.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: D
With 4 remaining choices: $\text{EV}_\text{guess} = \frac{6}{4} = 1.5$ vs. $\text{EV}_\text{blank} = 1.5$. Both have same expected value.
Problem 10
Tags: Expected Value · source: AMC10 2020 #22
You’re stuck on a problem with 5 answer choices. You can eliminate 3 choices. Should you guess or skip?
A) Guess
B) Skip
C) Need more information
D) Both have same expected value
E) Cannot be determined
Strategic Analysis
Calculation: 2 choices remaining. $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
Expected Value: 1 choice remaining, EV = 6 points.
Answer & Solution
Answer: A
With 2 remaining choices: $\text{EV}_\text{guess} = \frac{6}{2} = 3$ vs. $\text{EV}_\text{blank} = 1.5$. Since $3 > 1.5$, guess.
📊 Expected Value Reference Table
| Eliminations | Remaining | EV of Guessing | Decision |
|---|---|---|---|
| 0 | 5 | 1.2 | Skip |
| 1 | 4 | 1.5 | Either |
| 2 | 3 | 2.0 | Guess |
| 3 | 2 | 3.0 | Guess |
| 4 | 1 | 6.0 | Guess |
Next: Three-Pass Method | Back: Strategy Drills