Probability

Definitions

For disjoint events, $A \cap B = \emptyset$

A - B = A \cap B^{c}

A Δ B = (A - B) \cup (B - A) = A \cup B - (A \cap B)

A Δ B = (A - B) \cup (B - A) = (A \cap B^{c}) \cup (B \cap A^{c}) = (A \cup (B \cap A^{c})) \cap (B^{c} \cup (B \cap A^{c})) = ((A \cup B) \cap (A \cup A^{c})) \cap ((B^{c} \cup B) \cap (B^{c} \cup A^{c})) = (A \cup B) \cap (B^{c} \cup A^{c}) = (A \cup B) \cap (A \cap B)^{c} = A \cup B - (A \cap B)

A \cap B = B \cap A A \cup B = B \cup A

(A \cap B) \cap C = A \cap (B \cap C) (A \cup B) \cup C = A \cup (B \cup C)

(A \cup B) \cap C = (A \cap C) \cup (B \cap C) (A \cap B) \cup C = (A \cup C) \cap (B \cup C)

(A \cup B)^{c} = A^{c} \cap B^{c} (A \cap B)^{c} = A^{c} \cup B^{c}

$0 \leq P (E) \leq 1$
$P (Ω) = 1$
If $E_{1}, E_{2}, \dots$ are mutually exclusive events, then $P (⋃_{i = 1}^{n} E_{i}) = \sum_{i = 1}^{n} P (E_{i})$

$P (A^{c}) = 1 - P (A)$
$P (\emptyset) = 0$
If $A \subset B$ , then $P (A) \leq P (B)$
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
- If $P (A \cup B) \geq 0$ , $P (A \cup B) \leq P (A) + P (B)$
- $P (A \cap B) \geq 1 - P (A^{c}) - P (B^{c})$

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )} P (A \cap B) = P (A ∣ B) P (B) = P (B ∣ A) P (A)

P (A \cap B \cap C) = P (A) P (B ∣ A) P (C ∣ (A \cap B))

P (A ∣ B) + P (A^{c} ∣ B) = \frac{P ( A \cap B )}{P ( B )} + \frac{P ( A ^{c} \cap B )}{P ( B )} = \frac{P ( B )}{P ( B )} = 1

P (A \cap B^{c}) = P (A) - P (A \cap B) = P (A) - P (A) P (B) = P (A) (1 - P (B)) = P (A) P (B^{c})

P (B ∣ A) = \frac{P ( A ∣ B ) P ( B )}{P ( A )} = \frac{P ( A ∣ B ) P ( B )}{P ( A ∣ B ) P ( B ) + P ( A ∣ B ^{c} ) P ( B ^{c} )}

P (A \cap B) = P (A ∣ B) P (B) = P (B ∣ A) (P (A))