The result follows from manipulating the conditional probability. By definition,
[tex]P(A_1\mid B)=\dfrac{P(A_1\cap B)}{P(B)}=\dfrac{P(B\mid A_1)P(A_1)}{P(B)}[/tex]
[tex]P(A_1\mid B)=\dfrac{0.05\cdot0.60}{P(B)}[/tex]
By the law of total probability,
[tex]P(B)=P(B\cap A_1)+P(B\cap A_2)=P(B\mid A_1)P(A_1)+P(B\mid A_2)P(A_2)[/tex]
So we have
[tex]P(A_1\mid B)=\dfrac{0.05\cdot0.60}{0.05\cdot0.60+0.10\cdot0.40}\approx0.4286[/tex]
