Question

A manufacturing facility is responsible for producing two components that are produced at states 1 (poor), 2 (below average), 3 (average), 4 (good), and 5 (excellent). For the final product (con- sisting of both components) to be acceptable the sum of the two states needs to be at least equal to 6. At the moment, the manufacturing processes are random, and hence each component is equally likely to be at any of the 5 states. Answer the following questions.

Required:
a. What is the probability that a final product is acceptable? What is the probability that a final product is unacceptable?
b. What is the probability that a final product is acceptable given that the first component is produced at a state of 2?
c. What is the probability that the second component was produced at a state of 3 given that the final product is acceptable?

Answer 1

a) i) P( final product is acceptable ) = 3/5

ii) P ( final product not acceptable ) = 2/5

B) P ( final product is acceptable | X =2 ) = 2/5

C) P ( Y = 3 | Final product is acceptable ) = 2/5

Explanation:

Given data :

5 states : 1 (poor), 2 (below average), 3 (average), 4 (good), and 5 (excellent)

Number of components produced = 2

X ( component 1 ) ∴ P(X) = 1/5

Y ( component 2 ) ∴ P(Y) = 1/5

Aim : Sum of Final product needs to be ≥ 6

A) Probability of a final product been acceptable and also unacceptable

P( final product is acceptable ) ; P(X + Y ≥ 6 )

= 1/5 ∑ P( Y ≥ 6 - x )

= 1/5 ( P( Y ≥ 5 ) + P( Y ≥ 4 ) + P( Y ≥ 3 ) + P( Y ≥ 2 ) + P( Y ≥ 1 )

= 1/5 ( 1/5 + 2/5 + 3/5 +4/5 + 1 )

= 3/5

P ( final product not acceptable ) = 1 - 3/5 = 2/5

B) P ( final product is acceptable | X =2 )

∴ P ( Y ≥ 4 )

= 2/5

C) P ( Y = 3 | Final product is acceptable )

∴ P ( X ≥ 3 ) ≠ P( Y ≥ 3 ) ( because they are not independent )

= 1 - 3/5 = 2/5.