Independence & The Multiplication Rule
Master the logic of compound probabilities. Learn when events influence each other, how to calculate "AND" probabilities, and strategies for solving complex scenarios.
Independent vs. Dependent Events
Definition
Two events are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. If E's occurrence changes the probability of F, the events are dependent.
Independent Example
Flipping a coin twice.
The result of the first flip (Heads) has zero impact on the second flip. The coin has no memory.
Dependent Example
Drawing 2 cards without replacement.
If you draw a King first, there are fewer Kings left for the second draw. The probability changes.
The Multiplication Rule
To find the probability that two independent events both occur (Event E AND Event F), multiply their individual probabilities.
Crucial Constraint
This simple multiplication rule ONLY applies if the events are independent. If they are dependent, you must use conditional probability (covered in Section 5.4).
Monte Carlo Roulette
The "At Least One" Probability
Calculating "at least one" success directly is often tedious because you have to add up many scenarios (1 success, 2 successes, etc.). It is much faster to calculate the probability of zero successes (none) and subtract from 1.
LogicLens Strategy: Independent Trials
If you have n independent trials, the probability of "none" is simply the probability of failure multiplied n times.
Adaptive Assessment
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