Independent Probabilities

Q: The questions based on the following scenario: Production line I of a factory works 60% of the time, while production line II works 70% of the time, independently of each other.

(c) What is the probability that precisely one of the lines operate?

A:

(a)  The probability that both lines operate can be expressed as:

P(line I operates ∩ line II operates)

Since these events are independent, we know:

P(line I operates ∩ line II operates) = P(line I operates)*P(line II operates)

= .6 * .7 = .42

(b)  The probability that both lines are stopped follows similar logic:

P(line I does not operate ∩ line II does not operate) = P(line I does not operate)*P(line II does not operate)

= .4 * .3 = .12

(c)  The probability that only one line operates means that we have to have the following scenario:

Line I operates and Line II does not or Line II operates and Line I does not =

P(Line I operates)*P(Line II does not) + P(Line II operates)*P(Line I does not) =

.6 * .3 + .7 * .4

.18 + .28 = .46