Independent solution

How to solve this Exponential Memorylessness question

Setup

Setup

Let T be the exponential future lifetime and write its survival function in terms of the unknown rate.

Pr⁡(T≥t)=e−λt\Pr(T\ge t)=e^{-\lambda t}

Model

Model

The memoryless property says the conditional survival from year 10 to year 25 depends only on the additional 15 years.

Pr⁡(T≥25∣T≥10)=Pr⁡(T≥15)=e−15λ=0.40\Pr(T\ge25\mid T\ge10)=\Pr(T\ge15)=e^{-15\lambda}=0.40

Compute

Compute

Express the 50-year survival as a power of the calibrated 15-year survival probability.

e−50λ=(e−15λ)50/15e^{-50\lambda}=\left(e^{-15\lambda}\right)^{50/15}
Pr⁡(T≥50)=0.4010/3=0.0471556032…\Pr(T\ge50)=0.40^{10/3}=0.0471556032\ldots

Answer

Answer

The 50-year survival probability rounds to 0.05 at the precision of the choices.

Pr⁡(T≥50)≈0.05(B)\boxed{\Pr(T\ge50)\approx0.05\quad\text{(B)}}