Question

In a manufactory, there are two types of spoilages. It is found that 5% of spoilages are due to transformer spoilage, 8% are due to line spoilage and 3% involve both problems. (a) Are the two types of spoilages independent or not? Justify your answer. (b) What is the probability that: i- line spoilage given that there is transformer spoilage ii- Transformer spoilage but not line spoilage. iii- Transformer spoilage given that there is no line spoilage iv- Neither transformer spoilage nor there is no line spoilage v- Either transformer spoilage or there is no line spoilage

Answer 1

Explanation:

Let us denote probability of spoilage as follows

Transformer spoilage = P( T ) ; line spoilage P ( L )

Both P ( T ∩ L ) .

Given

P( T ) = .05

P ( L ) = .08

P ( T ∩ L ) = .03

a )

For independent events

P ( T ∩ L ) = P( T ) x P ( L )

But .03 ≠ .05 x .08

So they are not independent of each other .

b )

i )

Probability of line spoilage given that there is transformer spoilage

P L/ T ) = P ( T ∩ L ) / P( T )

= .03 / .05

= 3 / 5 .

ii )

Probability of transformer spoilage but not line spoilage.

P( T ) - P ( T ∩ L )

.05 - .03

= .02

iii )Probability of transformer spoilage given that there is no line spoilage

[ P( T ) - P ( T ∩ L ) ] / 1 - P ( L )

= .02 / 1 - .08

= .02 / .92

= 1 / 49.

i v )

Neither transformer spoilage nor there is no line spoilage

= 1 - P ( T ∪ L )

1 - [ P( T ) + P ( L ) - P ( T ∩ L ]

= 1 - ( .05 + .08 - .03 )

= 0 .9