Independent solution

How to solve this Conditional Probability question

Setup

Setup

Integrate the density to obtain its cumulative distribution function.

F(x)=∫0x2e−2t dt=1−e−2xF(x)=\int_0^x2e^{-2t}\,dt=1-e^{-2x}

Model

Model

The smaller-threshold event is a subset of the conditioning event, so their intersection is the smaller event.

Pr⁡(X≤0.5∣X≤1)=F(0.5)F(1)\Pr(X\le0.5\mid X\le1)=\frac{F(0.5)}{F(1)}

Compute

Compute

Evaluate the cdf at both thresholds and form the ratio.

F(0.5)=1−e−1F(0.5)=1-e^{-1}
F(1)=1−e−2F(1)=1-e^{-2}
1−e−11−e−2=0.7310585786\frac{1-e^{-1}}{1-e^{-2}}=0.7310585786

Answer

Answer

The conditional probability rounds to 0.731.

0.731(D)\boxed{0.731\quad\text{(D)}}