Compound Probability

Conditional Probability

Condition on event B by dividing the joint by the conditioning probability:

P(U|V) = P(U∩V) / P(V)

for events where P(V) > 0.

Example

Divide the joint probability by the condition probability.

highlighted = computed this step

Step 1 — Condition probability

Compute the condition probability.

P(B)=3/6=12P(B)= 3 / 6 = \hlmath{\frac{1}{2}}

Step 2 — Joint probability

Compute the joint probability.

P(A∩B)=2/6=13P(A\cap B)= 2 / 6 = \hlmath{\frac{1}{3}}

Step 3 — Divide by condition

Divide the joint probability by the condition.

P(A∣B)=P(A∩B)P(B)=13/12=23P(A|B)=\frac{P(A\cap B)}{P(B)}= \frac{1}{3} / \frac{1}{2} = \hlmath{\frac{2}{3}}

Step 4 — Result

State the conditional probability.

P(A∣B)=23P(A|B)= \hlmath{\frac{2}{3}}
conditional-probability P(U|V) = P(U∩V) / P(V)