# Union Probability: Device Failure Let's break down the problem step by step. ## Given Probabilities - Probability component A fails: \( P(A) = .01 \) - Probability component B fails: \( P(B) = .03 \) - Probability component C fails: \( P(C) =.04 \) - All failures are **independent**. The device **fails** if **any** of A, B, or C fails (union of events). ## Step 1: Probability Device Fails We want: \[ P(\text{device fails}) = P(A \cup B \cup C) \] For independent events: \[ P(A \cup B \cup C) = 1 - P(\overline{A} \cap \overline{B} \cap \overline{C}) \] where \(\overline{A}\) means "A does NOT fail" (A works). ## Step 2: Probability All Components Work Calculate: - Probability A works: \( P(\overline{A}) = 1 - .01 = .99 \) - Probability B works: \( P(\overline{B}) = 1 - .03 = .97 \) - Probability C works: \( P(\overline{C}) = 1 - .04 = .96 \) Since independent: \[ P(\text{All work}) = .99 \times .97 \times .96 \] Calculate stepwise: - \(.99 \times .97 = .9603\) - \(.9603 \times .96 = .921888\) So, \[ P(\text{All work}) = .921888 \] ## Step 3: Probability Device Works Since the device works only if **all** components work: \[ P(\text{device works}) = P(\overline{A} \cap \overline{B} \cap \overline{C}) = .921888 \] ## Step 4: Final Answers - **Probability device works:** \(\boxed{.9219}\) (rounded to 4 decimal places) - **Probability device fails:** \(1 - .921888 = .078112\) (rounded: \(\boxed{.0781}\)) --- ## Summary Table | Event | Probability | |----------------|-------------| | Device works | .9219 | | Device fails | .0781 | --- **Key Point:** To find the probability the device works (none fail), multiply the probabilities that each component works. For independent components, the union probability is \(1 - P(\text{all work})\).