Question

For mutually exclusive events r1​, r2​, and r3​, we have ​p(r1​) = 0.05​,​ p(r2​) = 0.6​, and ​p(r3​) = 0.35. ​also, p( q | r 1 (=0.6​, p (q | r 2 )=0.3​, and p ( q | r 3 ) = 0.6. find p ( r1 | q ).

Answer 1

From the definition of conditional probability:

$P(R_1\mid Q)=\dfrac{P(R_1\cap Q)}{P(Q)}$

By the law of total probability,

$P(Q)=P(Q\cap R_1)+P(Q\cap R_2)+P(Q\cap R_3)$

$P(Q)=P(Q\mid R_1)P(R_1)+P(Q\mid R_2)P(R_2)+P(Q\mid R_3)P(R_3)$

$P(Q)=0.42$

Since

$P(R_1\cap Q)=P(Q\mid R_1)P(R_1)$

we end up with

$P(R_1\mid Q)=\dfrac{0.03}{0.42}\approx0.0714$