Independent solution

How to solve this Independence question

Setup

Setup

Compute the probability that the first exponential time is at most 3 and the second independent exponential time is at most 2.

P(G≤3)=1−e−3/6=1−e−1/2P(G\le3)=1-e^{-3/6}=1-e^{-1/2}
P(B≤2)=1−e−2/3P(B\le2)=1-e^{-2/3}

Model

Model

Convert each deadline to its exponential cumulative probability. Independence makes the joint probability their product.

P(G≤3,B≤2)=P(G≤3)P(B≤2)P(G\le3,B\le2)=P(G\le3)P(B\le2)

Compute

Compute

Multiplying the two within-deadline probabilities gives 0.1915; expanding the product gives the expression displayed in choice C.

(1−e−1/2)(1−e−2/3)=0.1915(1-e^{-1/2})(1-e^{-2/3})=0.1915

Answer

Answer

The required joint probability is approximately 0.1915, corresponding to choice C.

0.1915(C)\boxed{0.1915\quad\text{(C)}}