Answer to Question #108848 in Statistics and Probability for Dhruv patel

Question

machine A,B and C all produce the same two parts X and Y.of all the parts produced,machine A produces 60% ,machine B produced 30% and machine C produced 10% in addition,40% of the same parts made by machine A are part X,50% of the parts made by machine B are part X,70% of the parts made by machine C are part X,A part produced by this company is randomly sample and is determined to be an X part. What is the probability that the part came from machine A.

Answer 1

The probability of production is given by,

$P(A) = 0.6,\\ P(B) = 0.3,\\ P(C) = 0.1.$

Part "X" made by each machine is given by,

$P(X | A) = 0.4,\\ P( X | B) = 0.5,\\ P(X | C ) = 0.7.$

Probability of producing part "X"

$P(X) = P(X | A) P(A) + P(X | B) P(B) + P(X | C) P(C)\\$

Substituting from above values gives,

$P(X)= 0.46$

From

$P(X | A) P(A) = P(A | X) P(X) \to P(A | X) = P(X | A) P(A) / P(X),$

$P(A | X) = 0.521$

Answer: $P(A | X) = 0.521$