If a machine is correctly set up,it produces | vedclass

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Question

(A) Let $B_{1}$ be the event that the machine is correctly set up and $B_{2}$ be the event that the machine is incorrectly set up.Let $A$ be the event

Vedclass · Accepted Answer

$0.9529$

Vedclass · Answer

(A) Let $B_{1}$ be the event that the machine is correctly set up and $B_{2}$ be the event that the machine is incorrectly set up.Let $A$ be the event that the machine produces $2$ acceptable items.Given:$P(B_{1}) = 0.8$$P(B_{2}) = 1 - 0.8 = 0.2$$P(A|B_{1}) = 0.9 	imes 0.9 = 0.81$$P(A|B_{2}) = 0.4 	imes 0.4 = 0.16$Using Bayes' Theorem,the probability that the machine is correctly set up given it produced $2$ acceptable items is:$P(B_{1}|A) = \frac{P(B_{1})P(A|B_{1})}{P(B_{1})P(A|B_{1}) + P(B_{2})P(A|B_{2})}$$P(B_{1}|A) = \frac{0.8 	imes 0.81}{(0.8 	imes 0.81) + (0.2 	imes 0.16)}$$P(B_{1}|A) = \frac{0.648}{0.648 + 0.032} = \frac{0.648}{0.680} = \frac{648}{680} = \frac{81}{85} \approx 0.9529$

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