Probability
The branch of mathematics that deals with the study of random phenomena and the quantification of uncertainty.
Key Concepts
- Sample Space: The set of all possible outcomes of a random experiment
- Event: A subset of the sample space
- Probability: A measure of the likelihood of an event occurring
- Conditional Probability: The probability of an event given that another event has occurred
- Independence: Events are independent if the occurrence of one does not affect the probability of the other
Probability Laws
- Axiom 1: For any event A, P(A) ≥ 0
- Axiom 2: P(S) = 1, where S is the sample space
- Axiom 3: For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B)
- Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Conditional Probability: P(A|B) = P(A ∩ B) / P(B), where P(B) > 0
- Multiplication Rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
Interactive Demonstrations
Dice Roll Simulator
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Each number has a 1/6 (≈16.7%) probability of occurring.
Coin Flip Simulator
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Heads and Tails each have a 1/2 (50%) probability of occurring.