Question

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 ?

• A. 3/4
• B. 1/2
• C. 1/32
• D. 1/4

Hint:

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.

Answer 1

The Correct Option is A

To solve the problem of determining the probability that the ship can complete the voyage, we need to consider the requirement that at least two of the engines (A, B, or C) must work. This means we should calculate the probability of any of the following scenarios:

• All three engines work.
• Exactly two engines work while one engine fails.

Let's denote:

• The probability that engine A works: \(P(\text{A works}) = 1 - \frac{1}{4} = \frac{3}{4}\)
• The probability that engine B works: \(P(\text{B works}) = 1 - \frac{1}{4} = \frac{3}{4}\)
• The probability that engine C works: \(P(\text{C works}) = 1 - \frac{1}{2} = \frac{1}{2}\)

Let's calculate the probability of each of these scenarios:

1. All three engines work:

The probability for this is:

\(P(\text{All work}) = P(\text{A works}) \times P(\text{B works}) \times P(\text{C works}) = \frac{3}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{9}{32}\)

1. Exactly two engines work:

The three possible cases are:

• Engines A and B work, and C fails:

The probability for this is:

\(P(\text{A and B work, C fails}) = \frac{3}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{9}{32}\)

• Engines A and C work, and B fails:

The probability for this is:

\(P(\text{A and C work, B fails}) = \frac{3}{4} \times \frac{1}{2} \times \frac{1}{4} = \frac{3}{32}\)

• Engines B and C work, and A fails:

The probability for this is:

\(P(\text{B and C work, A fails}) = \frac{1}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{3}{32}\)

Adding the probabilities of all engines working and exactly two engines working gives us the total probability that the ship can complete the voyage:

\(P(\text{Ship completes}) = \frac{9}{32} + \frac{15}{32} = \frac{24}{32} = \frac{3}{4}\)

Hence, the probability that the ship can complete the voyage is 3/4.