Independent solution

How to solve this Conditional Probability question

Setup

Setup

Let T be the total completion time. Standardize T using the given mean and standard deviation.

T∼N(7.5,1.52),Z=T−7.51.5T\sim N(7.5,1.5^2),\qquad Z=\frac{T-7.5}{1.5}

Model

Model

The requested event lies inside the conditioning event, so conditional probability is the interval probability divided by the upper-half probability.

Pr⁡(T≤8∣T>7.5)=Pr⁡(7.5<T≤8)Pr⁡(T>7.5)\Pr(T\le8\mid T>7.5)=\frac{\Pr(7.5<T\le8)}{\Pr(T>7.5)}
Pr⁡(T>7.5)=12\Pr(T>7.5)=\frac12

Compute

Compute

The upper endpoint is one-third of a standard deviation above the mean.

8−7.51.5=13\frac{8-7.5}{1.5}=\frac13
Pr⁡(T≤8∣T>7.5)=Φ(1/3)−Φ(0)1/2\Pr(T\le8\mid T>7.5)=\frac{\Phi(1/3)-\Phi(0)}{1/2}
=2Φ(1/3)−1=0.2611173196…=2\Phi(1/3)-1=0.2611173196\ldots

Answer

Answer

The conditional probability is approximately 0.26.

0.26(B)\boxed{0.26\quad\text{(B)}}