🔢 What is Probability?
Probability is the branch of mathematics that deals with uncertainty. It helps us quantify how likely something is to happen.
Basic formula:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Example: Rolling a die and getting a 4 → P(4) = 1 / 6
🧠 Key Terms
- Experiment: A process that leads to an outcome (e.g., tossing a coin)
- Outcome: A single result of an experiment (e.g., heads)
- Sample Space (S): The set of all possible outcomes (e.g., {Heads, Tails})
- Event: A subset of the sample space (e.g., getting a tail)
📚 Types of Probability
- Theoretical Probability: Based on reasoning (e.g., fair die → P(2) = 1/6)
- Experimental Probability: Based on actual trials or experiments
P(E) = Number of times E occurs / Total trials - Subjective Probability: Based on intuition or experience (e.g., "I feel there's a 70% chance of rain")
📐 Basic Probability Rules
- Rule 1: 0 ≤ P(E) ≤ 1
- Rule 2: The sum of probabilities of all outcomes in the sample space is 1
- Complement Rule:
P(not A) = 1 - P(A) - Addition Rule (Mutually Exclusive):
P(A or B) = P(A) + P(B) - Addition Rule (General):
P(A or B) = P(A) + P(B) - P(A and B)
🔗 Conditional Probability & Independence
Conditional Probability: Probability of A happening given B has happened
P(A | B) = P(A and B) / P(B)
Independent Events: Events where the outcome of one doesn't affect the other
If A and B are independent: P(A and B) = P(A) × P(B)
📊 Tree Diagrams & Tables
Tree diagrams and two-way tables can help visualize multi-step probability problems, especially when dealing with dependent events.
🧮 Permutations & Combinations
Permutations: Arrangements where order matters
P(n, r) = n! / (n - r)!
Combinations: Selections where order doesn’t matter
C(n, r) = n! / (r!(n - r)!)
Example: How many 3-letter passwords from A, B, C, D?
- Permutations: 24
- Combinations: 4
🌍 Applications of Probability
- Weather forecasting
- Medical testing (false positives/negatives)
- Games and gambling
- Machine learning and statistics
- Finance and risk analysis
- Quality control and reliability
📝 Practice Questions
- What is the probability of getting an even number when rolling a 6-sided die?
- If you flip 2 coins, what is the probability of getting at least one head?
- What's the chance of drawing a red card from a standard 52-card deck?
- If P(A) = 0.6 and P(B) = 0.5, and A and B are independent, what is P(A and B)?