Question

A safety device has two independent critical components $X_1$ and $X_2$. Failure of any one or both causes an accident. Failure probabilities: $P(X_1)=0.2$, $P(X_2)=0.1$. Find the probability of an accident (rounded to two decimals).

Hint:

For independent components, use $P(\text{system fails}) = 1 - P(\text{all components work})$.

Answer 1

Correct Answer: 0.28

An accident occurs if:
- $X_1$ fails, or
- $X_2$ fails, or
- both fail.
Since the failures are independent: \[ P(\text{no accident}) = P(X_1\ \text{works})\cdot P(X_2\ \text{works}) \] \[ = (1-0.2)(1-0.1) = 0.8 \cdot 0.9 = 0.72. \] Thus accident probability: \[ P(\text{accident}) = 1 - 0.72 = 0.28. \] \[ \boxed{0.28} \]