Independent solution

How to solve this Conditional Probability question

Answer in brief

Integrating the density gives survival S(t)=e^(-t)(1+t). Conditional on lasting beyond one year, the probability of ending by year two is 1-S(2)/S(1)=1-3/(2e)=0.448181, so choice C is correct.

Setup

Setup

Integrate the density tail to obtain the survival function.

S(t)=∫t∞ue−u du=e−t(1+t)S(t)=\int_t^\infty ue^{-u}\,du=e^{-t}(1+t)

Model

Model

Express the conditional interval probability using survival at the two endpoints.

Pr⁡(T≤2∣T≥1)=S(1)−S(2)S(1)\Pr(T\le2\mid T\ge1)=\frac{S(1)-S(2)}{S(1)}

Compute

Compute

Evaluate and simplify the survival ratio.

S(1)=2e−1,S(2)=3e−2S(1)=2e^{-1},\qquad S(2)=3e^{-2}
Pr⁡(T≤2∣T≥1)=1−3e−22e−1\Pr(T\le2\mid T\ge1)=1-\frac{3e^{-2}}{2e^{-1}}
=1−32e=0.4481808382…=1-\frac{3}{2e}=0.4481808382\ldots

Answer

Answer

The conditional probability rounds to 0.448.

0.448(C)\boxed{0.448\quad\text{(C)}}