A ship sets off on a voyage with three engines, labelled A, B, and C, which work independently. The ship can complete the voyage only if at least two of these engines keep working. The probability that engine A breaks down is 1/4, that engine B breaks down is 1/4, and that engine C breaks down is 1/2. What is the probability that the ship can complete the voyage ?
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An alternative way is to use the complement:
\[ P(\text{Complete}) = 1 - P(\text{0 engines work}) - P(\text{Exactly 1 engine works}) \]
Often, direct computation is equally fast when the number of successful cases is small.
• Step 1 : Understanding the Question:
We are given a system with three independent engines, A, B, and C.
The breakdown probabilities of the three engines are given.
The ship completes the voyage if at least two engines keep working.
We need to find the total probability of a successful voyage.
• Step 2 : Key Formula or Approach:
The probability of an engine working is \(1 - P(\text{breakdown})\).
The voyage is successful if:
1. All three engines work.
2. Exactly two engines work.
Since these scenarios are mutually exclusive, the total probability is the sum of the individual probabilities.
• Step 3 : Detailed Explanation:
Let us define the probabilities of working for each engine:
- Engine A:
\[ P(A_{\text{break}}) = \frac{1}{4} \implies P(A_{\text{work}}) = 1 - \frac{1}{4} = \frac{3}{4} \]
- Engine B:
\[ P(B_{\text{break}}) = \frac{1}{4} \implies P(B_{\text{work}}) = 1 - \frac{1}{4} = \frac{3}{4} \]
- Engine C:
\[ P(C_{\text{break}}) = \frac{1}{2} \implies P(C_{\text{work}}) = 1 - \frac{1}{2} = \frac{1}{2} \]
Since the engines work independently, we can multiply their individual probabilities to find the probability of simultaneous occurrences.
Let us calculate the probabilities of the successful cases:
- Case 1: All three engines work:
\[ P(\text{A work, B work, C work}) = P(A_{\text{work}}) \cdot P(B_{\text{work}}) \cdot P(C_{\text{work}}) \]
\[ P(\text{All 3 work}) = \frac{3}{4} \cdot \frac{3}{4} \cdot \frac{1}{2} = \frac{9}{32} \]
- Case 2: Exactly two engines work. This can happen in three ways:
1. A and B work, C breaks down:
\[ P(A_{\text{work}} \cap B_{\text{work}} \cap C_{\text{break}}) = \frac{3}{4} \cdot \frac{3}{4} \cdot \frac{1}{2} = \frac{9}{32} \]
2. A and C work, B breaks down:
\[ P(A_{\text{work}} \cap B_{\text{break}} \cap C_{\text{work}}) = \frac{3}{4} \cdot \frac{1}{4} \cdot \frac{1}{2} = \frac{3}{32} \]
3. B and C work, A breaks down:
\[ P(A_{\text{break}} \cap B_{\text{work}} \cap C_{\text{work}}) = \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{32} \]
Summing up all the mutually exclusive successful scenarios:
\[ P(\text{Voyage Completed}) = \frac{9}{32} + \frac{9}{32} + \frac{3}{32} + \frac{3}{32} \]
\[ P(\text{Voyage Completed}) = \frac{24}{32} = \frac{3}{4} \]
• Step 4 : Final Answer:
The probability that the ship completes the voyage is \(\frac{3}{4}\).
This corresponds to option (A).
