🎲 Probability & EV Snippets
Master essential probability and expected value calculations through quick practice snippets. These focused exercises will build speed and accuracy in probability problems.
🎯 Core Probability Concepts
Basic Probability
- Definition: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
- Range: $0 \leq P(A) \leq 1$
- Complement: $P(A^c) = 1 - P(A)$
- Union: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Intersection: $P(A \cap B) = P(A) \cdot P(B)$ (if independent)
Conditional Probability
- Definition: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
- Multiplication rule: $P(A \cap B) = P(A|B) \cdot P(B)$
- Bayes’ theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$
Independence
- Definition: Events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$
- Equivalent: $P(A|B) = P(A)$ or $P(B|A) = P(B)$
- Multiple events: $P(A_1 \cap A_2 \cap \ldots \cap A_n) = P(A_1) \cdot P(A_2) \cdot \ldots \cdot P(A_n)$
⚡ Expected Value Mastery
Basic Expected Value
- Definition: $E[X] = \sum x \cdot P(X = x)$
- Linearity: $E[X + Y] = E[X] + E[Y]$
- Constants: $E[cX] = c \cdot E[X]$
- Independence: $E[XY] = E[X] \cdot E[Y]$ (if independent)
Indicator Variables
- Definition: $I_A = \begin{cases} 1 & \text{if event A occurs} \ 0 & \text{if event A does not occur} \end{cases}$
- Properties: $E[I_A] = P(A)$, $E[I_A^2] = P(A)$, $E[I_A I_B] = P(A \cap B)$
- Applications: Counting problems, probability calculations
Common Distributions
- Uniform: $P(X = k) = \frac{1}{n}$ for $k = 1, 2, \ldots, n$
- Bernoulli: $P(X = 1) = p$, $P(X = 0) = 1-p$, $E[X] = p$
- Binomial: $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$, $E[X] = np$
- Geometric: $P(X = k) = (1-p)^{k-1}p$, $E[X] = \frac{1}{p}$
🎯 Practice Drills
5-Minute Sprint: Basic Probability
Target: 20 problems in 5 minutes (90%+ accuracy)
- What is the probability of rolling a 6 on a fair die?
- What is the probability of rolling an even number on a fair die?
- What is the probability of rolling a number greater than 4 on a fair die?
- What is the probability of rolling a 1 or 2 on a fair die?
- What is the probability of rolling a number less than 3 on a fair die?
5-Minute Sprint: Conditional Probability
Target: 15 problems in 5 minutes (85%+ accuracy)
- If you roll a die and get an even number, what is the probability it’s a 6?
- If you roll a die and get a number greater than 3, what is the probability it’s a 6?
- If you roll a die and get a number less than 5, what is the probability it’s a 4?
- If you roll a die and get a number greater than 2, what is the probability it’s a 5?
- If you roll a die and get a number less than 6, what is the probability it’s a 3?
5-Minute Sprint: Expected Value
Target: 15 problems in 5 minutes (85%+ accuracy)
- What is the expected value of rolling a fair die?
- What is the expected value of rolling two fair dice?
- What is the expected value of flipping a fair coin?
- What is the expected value of flipping two fair coins?
- What is the expected value of rolling a die and flipping a coin?
🔢 Advanced Probability Techniques
Principle of Inclusion-Exclusion
- Two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Three events: $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$
- General formula: Alternating sum of intersections
Complementary Counting
- “At least” problems: $P(\text{at least one}) = 1 - P(\text{none})$
- “At most” problems: $P(\text{at most one}) = 1 - P(\text{more than one})$
- “Not all” problems: $P(\text{not all}) = 1 - P(\text{all})$
Symmetry
- Equal probability: When outcomes are equally likely
- Symmetric problems: When problem has symmetric structure
- Geometric problems: When shapes are symmetric
- Combinatorial problems: When arrangements are symmetric
🎯 Advanced Practice Drills
5-Minute Sprint: PIE and Complements
Target: 12 problems in 5 minutes (80%+ accuracy)
- In a class of 30 students, 18 like math and 15 like science. If 8 like both, what is the probability a random student likes at least one?
- What is the probability of getting at least one head in 3 coin flips?
- What is the probability of getting at least one 6 in 3 die rolls?
- What is the probability of getting at least one even number in 2 die rolls?
- What is the probability of getting at least one head in 2 coin flips?
5-Minute Sprint: Indicator Variables
Target: 10 problems in 5 minutes (80%+ accuracy)
- What is the expected number of heads in 5 coin flips?
- What is the expected number of 6’s in 10 die rolls?
- What is the expected number of even numbers in 8 die rolls?
- What is the expected number of heads in 12 coin flips?
- What is the expected number of 1’s in 6 die rolls?
📊 Progress Tracking
Accuracy Targets
- Basic probability: 90%+ accuracy
- Conditional probability: 85%+ accuracy
- Expected value: 85%+ accuracy
- Advanced techniques: 80%+ accuracy
Speed Targets
- Basic probability: 4 problems per minute
- Conditional probability: 3 problems per minute
- Expected value: 3 problems per minute
- Advanced techniques: 2 problems per minute
Weekly Goals
- Week 1: Master basic probability and expected value
- Week 2: Add conditional probability, maintain accuracy
- Week 3: Add advanced techniques, increase speed
- Week 4: Master all areas, optimize speed
⚡ Quick Reference
Essential Probability Formulas:
- Basic: $P(A) = \frac{\text{favorable}}{\text{total}}$
- Complement: $P(A^c) = 1 - P(A)$
- Union: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Conditional: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
Essential Expected Value Formulas:
- Basic: $E[X] = \sum x \cdot P(X = x)$
- Linearity: $E[X + Y] = E[X] + E[Y]$
- Indicator: $E[I_A] = P(A)$
- Binomial: $E[X] = np$
Common Distributions:
- Uniform: $E[X] = \frac{n+1}{2}$
- Bernoulli: $E[X] = p$
- Binomial: $E[X] = np$
- Geometric: $E[X] = \frac{1}{p}$
Practice Schedule:
- Daily: 10 minutes of probability practice
- Focus areas: Work on your weakest skills
- Progressive difficulty: Increase complexity over time
- Time pressure: Practice under time constraints
- Accuracy first: Speed comes with accuracy
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