Imagine you and Janice are the only ones working as telemarketers for a large cooperation. You make 70% of the calls and each call takes you Exponential amount of time with parameter λ1 = 2 hrs-1 . Janice, on the other hand, makes the remaining 30% of total calls. Each call takes her Exponential amount of time with parameter λ2 = 4 hrs-1 . A certain call was made 30 minutes ago and it is still ongoing. What is the probability that you made this call?

The probability that you made this call is 0.8639.

Explanation:

Parameter for you = λ1 = 2

⇒ β1 = 1 / λ1 = 1 / 2 = 0.5

Parameter for Janice = λ2 = 4

⇒ β2 = 1 / λ2 = 1 / 4 = 0.25

Probability that you made a for 0.5 hours or more:

Probability that Janice made a for 0.5 hours or more:

Overall probability that a call was made for 0.5 hours of more

= (0.7 x 0.3679) + (0.3 x 0.1353)

= 0.2981

Probability that you made the given call

= 0.7 x 0.3679 / 0.2981

= 0.8639

expression #3 is not equivalent. the others are.