Compound Probability

Multiplication Rule for Dependent Events

When events are not independent, use conditional probability:

P(R∩R) = P(R) · P(R|R)

denominators update to remaining sample outcomes.

Example

Use the updated conditional probability after the first event.

highlighted = computed this step

Step 1 — First draw

Compute the first draw probability.

P(R)=3/5=35P(R)= 3 / 5 = \hlmath{\frac{3}{5}}

Step 2 — Conditional draw

Update the denominator after the first draw.

P(R∣R)=2/4=12P(R|R)= 2 / 4 = \hlmath{\frac{1}{2}}

Step 3 — Multiply

Multiply by the conditional probability.

P(R∩R)=35⋅12=310P(R\cap R)= \frac{3}{5} \cdot \frac{1}{2} = \hlmath{\frac{3}{10}}

Step 4 — Result

State the dependent joint probability.

P(R∩R)=310P(R\cap R)= \hlmath{\frac{3}{10}}
multiplication-rule-dependent P(R∩R) = P(R) · P(R|R)