Probability

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An event is any subset of a sample space S. It represents a collection of outcomes of a random experiment. For example, if we toss a coin twice and S = {HH, HT, TH, TT}, then getting exactly one head …

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Impossible Event: The empty set φ, representing an event that cannot occur. Example: Getting a 7 when rolling a standard die. Sure Event: The whole sample space S, representing an event that always o…

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Simple Event: An event containing only one sample point. Example: {HH} when tossing two coins. Compound Event: An event containing more than one sample point. Example: Getting at least one head when …

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Axiom 1: For any event E, P(E) ≥ 0 (probability is non-negative) Axiom 2: P(S) = 1 (probability of sample space is 1) Axiom 3: If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F)…

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The complementary event A' (or 'not A') consists of all outcomes in the sample space S that are not in A. A' = S - A = {ω : ω ∈ S and ω ∉ A} Formula: P(A') = 1 - P(A)…

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'A or B': The union A ∪ B = {ω : ω ∈ A or ω ∈ B}, representing outcomes in either A or B or both. 'A and B': The intersection A ∩ B = {ω : ω ∈ A and ω ∈ B}, representing outcomes common to both A and…

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'A but not B' represents the event where A occurs but B does not occur. Set notation: A - B = A ∩ B' This includes all outcomes that are in A but not in B.

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P(E) = n(E)/n(S) = Number of outcomes favorable to E / Total number of possible outcomes This formula applies when all outcomes in the sample space are equally likely to occur.