Multiplication Rule

139. Multiplication Rule

For $𝑛$ events:

𝑃 (⋂_{𝑖 = 1}^{𝑛} 𝐴_{𝑖}) = \prod_{𝑖 = 1}^{𝑛} 𝑃 (𝐴_{𝑖} | ⋂_{𝑗 = 1}^{𝑖 - 1} 𝐴_{𝑗})

2 events:

𝑃 (𝐴_{1} \cap 𝐴_{2}) = 𝑃 (𝐴_{1}) 𝑃 (𝐴_{2} | 𝐴_{1})

3 events:

𝑃 (𝐴_{1} \cap 𝐴_{2} \cap 𝐴_{3}) = 𝑃 (𝐴_{1}) 𝑃 (𝐴_{2} | 𝐴_{1}) 𝑃 (𝐴_{3} | 𝐴_{1} \cap 𝐴_{2})

4 events:

𝑃 (𝐴_{1} \cap 𝐴_{2} \cap 𝐴_{3} \cap 𝐴_{4}) = 𝑃 (𝐴_{1}) 𝑃 (𝐴_{2} | 𝐴_{1}) 𝑃 (𝐴_{3} | 𝐴_{1} \cap 𝐴_{2}) 𝑃 (𝐴_{4} | 𝐴_{1} \cap 𝐴_{2} \cap 𝐴_{3})