 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/14/a/v/solution

The condition

X > x_{0}

x_{1}

and

x_{2}

. Their combined good-ball joint probability is

p (g_{1} + g_{2})

, and their total probability is

p (g_{1} + g_{2}) + (1 - p) (d_{1} + d_{2})

. Thus

P (G ∣ X > x_{0}) = \frac{p ( g _{1} + g _{2} )}{p ( g _{1} + g _{2} ) + ( 1 - p ) ( d _{1} + d _{2} )} = \frac{p ( 1 - g _{0} )}{p ( 1 - g _{0} ) + ( 1 - p ) ( 1 - d _{0} )} .

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