Conditional probability

Conditional probability allows the investigation of how the knowledge of one event occurring effects the knowledge of the other one. Given a probability model $(𝜉, F, ℙ)$ , and two events $𝐴, 𝐵 \in F$ , the conditional probability of $𝐴$ given $𝐵$ is #m/def/prob

ℙ (𝐴 ∣ 𝐵) = ℙ (𝐴 \cap 𝐵) ℙ (𝐵)

unless $ℙ (𝐵) = 0$ , in which case $ℙ (𝐴 ∣ 𝐵) = 0$ .¹ The function $ℙ (\cdot ∣ 𝐴)$ forms a probability measure on the same space $(𝜉, F)$ as $ℙ$ .

Properties

$ℙ (𝐴 \cap 𝐵) = ℙ (𝐵) ℙ (𝐴 ∣ 𝐵) = ℙ (𝐴) ℙ (𝐵 ∣ 𝐴)$
$ℙ (𝐴 ∣ 𝐵) = ℙ (𝐵 ∣ 𝐴) ℙ (𝐴) ℙ (𝐵)$
$ℙ (𝐴 ∣ 𝐵) ℙ (𝐴 𝑐 ∣ 𝐵) = ℙ (𝐵 ∣ 𝐴) ℙ (𝐵 ∣ 𝐴 𝑐) ℙ (𝐴) ℙ (𝐴 𝑐)$
Let ${𝐴 𝑖} 𝑛 𝑖 = 1$ partition $𝜉$ . Then $ℙ (𝐵) = 𝑛 \sum 𝑖 = 1 ℙ (𝐵 ∣ 𝐴 𝑖) ℙ (𝐴 𝑖)$
$ℙ (𝐴 ∣ 𝐵 \cap 𝐸) = ℙ (𝐵 ∣ 𝐴 \cap 𝐸) ℙ (𝐴 ∣ 𝐸) ℙ (𝐵 ∣ 𝐸)$

Conditional probability

Properties

See also

Footnotes