I have a question from A First Course in Probability by Sheldon Ross.
Question:
Consider two components and three types of shocks. A type 1 shock causes component 1 to fail, a type 2 shock causes component 2 to fail, and a type 3 shock causes both components 1 and 2 to fail. The times until shocks 1, 2, and 3 occur are independent exponential random variables with respective rates λ1, λ2, and λ3. Let Xi denote the time at which component i fails, i = 1, 2. The random variables X1,X2 are said to have a joint bivariate exponential distribution. Find P{X1 > s, X2 > t}.
The given solution is:
However, the solution given seems to suggest that all the types of shocks would need to occur. But shouldn't P{X1 >s, X2 > t} be the probability P{T1 >s, T2 > t} + P{T3 > max(s,t)} - P{T1 >s, T2 > t}P{T3 > max(s,t)}, since all the shocks are independent, and a type 3 shock would automatically cause both components to fail. Someone please confirm my reasoning or please tell me why my reasoning is flawed.
$\{X_1 > s, X_2 > t\}$ is "the first component is still working at time $s$, and the second component is still working at time $t$."
Thus the first Type 1 shock (if it comes at all) must come after time $s$ AND the first Type 2 shock must come after time $t$ AND the first Type 3 shock must come after $\max\{s,t\}$.
What you have written is the probability of $\{T_1 > s, T_2 > t\} \cup \{T_3 > \max\{s,t\}\}$. The union can be interpreted as an "OR", and you can see that it is not the same as the above paragraph which has three ANDs.