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Basic

P = favorable/total

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A AND B

Joint probability

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A OR B

Union of events

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P(A|B)

Conditional prob.

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Complement

P(not A)

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Binomial

n trials, k successes

Probability

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0 (Impossible)0.51 (Certain)

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Choose (r)

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📐 Probability Formulas

Basic Probability

P(A) = favorable outcomes / total outcomes

Range: 0 ≤ P(A) ≤ 1
P(impossible) = 0
P(certain) = 1
P(A) + P(not A) = 1

AND (Intersection)

Independent events: P(A∩B) = P(A) × P(B)
Dependent events: P(A∩B) = P(A) × P(B|A)

Example: P(head AND 6) = 0.5 × (1/6) = 1/12

OR (Union)

P(A∪B) = P(A) + P(B) − P(A∩B)

Mutually exclusive: P(A∪B) = P(A) + P(B)

Example: P(Heart OR Face card)
= 13/52 + 12/52 − 3/52 = 22/52 ≈ 42.3%

Conditional Probability

P(A|B) = P(A∩B) / P(B)

Read as: "Probability of A given B has occurred"

Bayes' Theorem:
P(A|B) = P(B|A) × P(A) / P(B)

Combinations & Permutations

Combination (order doesn't matter):
C(n,r) = n! / (r! × (n−r)!)

Permutation (order matters):
P(n,r) = n! / (n−r)!

With repetition: nʳ arrangements

Binomial Distribution

P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ

Where: n = trials, k = successes, p = prob success

Mean = n×p
Std Dev = √(n×p×(1-p))

Example: P(exactly 3 heads in 10 flips)
= C(10,3) × 0.5³ × 0.5⁷
= 120 × 0.125 × 0.0078125 = 11.72%

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