Probability

2021-01-08 Probability shark 1 min read

Attributes:

$P( \varnothing) = 0$

$0 \leq P(A) \leq 1$

Complement: $P(\overline{A}) = 1 - P(A)$

Union:

For two events, A and B, $P (A \cup B) = P(A) + P(B) - P(A \cap B)$

For three events A, B, C

$P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$

If A and B are mutually exclusive, then $P(A \cup B) = P(A) + P(B)$

Particial Theorem

If $B_1, B_2, ..., B_i$ are partitions of a sample space, Then

$P(A) = \sum_{i=1}^m P(A \cap B_i) = \sum_{i=1}^m P(A | B_i) P(B_i)$

If just $B$ and $\overline{B}$ are partitions, then

$P(A) = P(A \cap B) + P(A \cap \overline{B}) = P(A | B)P(B) + P(A | \overline{B}) P(\overline{B})$

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