Mathematics
Probability
Probability is the chance an event happens, between 0 (never) and 1 (certain). The basics: P(A) = favorable outcomes / total outcomes. Things get interesting with combined events and conditional probabilities.
How to Approach Probability
- 01
Count the sample space
List or count every possible outcome. For a fair die there are 6 outcomes; for two dice there are 36.
- 02
Count favorable outcomes
How many of those outcomes match the event you care about? Be careful with 'or' (add, then subtract overlap) and 'and' (multiply for independent events).
- 03
Divide & adjust
P(event) = favorable / total. For conditional probability, restrict the sample space to the given condition first.
Frequently Asked Questions
What's the difference between independent and dependent?
Independent events don't influence each other (two coin flips). Dependent events do (drawing cards without replacement). For independent: P(A and B) = P(A)·P(B).
What is Bayes' theorem?
P(A|B) = P(B|A)·P(A) / P(B). It lets you reverse a conditional probability — given P(B|A), find P(A|B).
What's expected value?
The long-run average outcome, weighted by probability: E[X] = Σ x·P(x). For a fair die: E = 3.5.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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