No Title

Proposition

Proposition
P(E₁|F)=P(E₁|E₂F)P(E₂|F)+P(E₁|E₂^cF)P(E₂^c|F)

If we define Q(E)=P(E|F), then Q(E) may be regarded as a probability function on the events of S. Hence all of the propositions previously proved for probabilities apply to it. For instance, we have

$Q(E_1\cup E_2)=Q(E_1)+Q(E_2)-Q(E_1E_2\vert F)$

or, equivalently,

$P(E_1\cup E_2\vert F)=P(E_1\vert F)+P(E_2\vert F)-P(E_1E_2\vert F)$

Also, if we define the conditional probability Q(E₁|E₂) by Q(E₁|E₂)=Q(E₁E₂)/Q(E₂), then we have

$Q(E_1)=Q(E_1\vert E_2)Q(E_2)+Q(E_1\vert E_2^c)Q(E_2^c)\qquad\qquad(5.1)$

Since

$\begin{array}{rcl} Q(E_1\vert E_2)&=&\displaystyle\frac{Q(E_1E_2)}{Q(E_2)} \\ \... ...(F)}}{\displaystyle\frac{P(E_2F)}{P(F)}} \\ \\ &=&P(E_1\vert E_2F) \end{array}$

we see that Equation (5.1) is equivalent to P(E₁|F)=P(E₁|E₂F)P(E₂|F)+P(E₁|E₂^cF)P(E₂^c|F)