How to Solve Continuous Random Variables — Univariate RV — Exponential distribution, conditional probability Questions on Exam P

An exponential waiting time has a survival function whose exponents scale linearly with elapsed time. Memorylessness also makes a conditional future interval depend only on the additional waiting time, not elapsed time. For an exponential variable \(T\), survival satisfies \(S_T(t)=e^{-\lambda t}\). Given \(\Pr(T\le 50)=0.20\), the survival probability at \(50\) hours is \(S_T(50)=1-0.20=0.80\). Therefore, survival to \(80\) hours is \(S_T(80)=S_T(50)^{\dfrac{80}{50}}=(0.80)^{\dfrac{80}{50}}=0.699752\). The probability that the interruption occurs within \(80\) hours is the complement of survival: \(\Pr(T\le 80)=1-S_T(80)=1-0.699752=0.3002\). Thus \(\Pr(T\leq80)=0.3002\approx0.3002\), which matches the correct option.