Independent solution

How to solve this Order Statistics question

Setup

Setup

The maximum is at most three exactly when all three independent losses are at most three.

M=max⁡(S,F,T)M=\max(S,F,T)
P(M≤3)=P(S≤3)P(F≤3)P(T≤3)P(M\le3)=P(S\le3)P(F\le3)P(T\le3)

Model

Model

Use the exponential cumulative probability for each loss and multiply the three probabilities by independence.

P(M>3)=1−(1−e−3)(1−e−2)(1−e−1.25)P(M>3)=1-(1-e^{-3})(1-e^{-2})(1-e^{-1.25})

Compute

Compute

The requested upper-tail event is the complement of the all-at-most-three event, giving approximately 0.4138.

P(M>3)=0.4140P(M>3)=0.4140

Answer

Answer

The probability that the maximum exceeds three is approximately 0.414, selecting choice E.

0.414(E)\boxed{0.414\quad\text{(E)}}