Independent solution

How to solve this Continuous Random Variables question

Setup

Setup

Normalize the density over its finite support.

1=∫050kt dt=1250k1=\int_0^{50}kt\,dt=1250k
k=11250k=\frac1{1250}

Model

Model

Write the conditional event as an interval inside the surviving population.

Pr⁡(T≤25∣T>20)=Pr⁡(20<T≤25)Pr⁡(T>20)\Pr(T\le25\mid T>20)=\frac{\Pr(20<T\le25)}{\Pr(T>20)}

Compute

Compute

Integrate the normalized density over the numerator and denominator ranges.

Pr⁡(20<T≤25)=∫2025t1250 dt=0.09\Pr(20<T\le25)=\int_{20}^{25}\frac{t}{1250}\,dt=0.09
Pr⁡(T>20)=∫2050t1250 dt=0.84\Pr(T>20)=\int_{20}^{50}\frac{t}{1250}\,dt=0.84
0.090.84=328=0.1071428571…\frac{0.09}{0.84}=\frac3{28}=0.1071428571\ldots

Answer

Answer

The conditional probability rounds to 0.11.

0.11(B)\boxed{0.11\quad\text{(B)}}