An aircraft has three engines A, B and C. The aircraft crashes if all the three engines fail. The probabilities of failure are 0.03, 0.02 and 0.05 for engines A, B and C respectively. What is the probability that the aircraft will not crash?

Options

A. 0.00003
B. `0.90`
C. 0.99997
D. 0.90307

Answer

To solve the problem, we need to find the probability that the aircraft will not crash, given the probabilities of failure for each engine.

### Step-by-Step Solution:

1. **Identify the probabilities of failure for each engine:**
- Probability of failure for Engine A, \( P(A) = 0.03 \)
- Probability of failure for Engine B, \( P(B) = 0.02 \)
- Probability of failure for Engine C, \( P(C) = 0.05 \)

2. **Calculate the probability that all engines fail:**
The aircraft crashes if all three engines fail. Therefore, we need to find the joint probability of all three engines failing:
\[ P(\text{Crash}) = P(A) \times P(B) \times P(C) \]
Substituting the values:
\[ P(\text{Crash}) = 0.03 \times 0.02 \times 0.05 \]

3. **Perform the multiplication:**
First, multiply \( 0.03 \) and \( 0.02 \):
\[ 0.03 \times 0.02 = 0.0006 \]
Now, multiply this result by \( 0.05 \):
\[ 0.0006 \times 0.05 = 0.00003 \]

4. **Calculate the probability that the aircraft will not crash:**
The probability that the aircraft does not crash is the complement of the probability that it crashes:
\[ P(\text{Not Crash}) = 1 - P(\text{Crash}) \]
Substituting the value we found:
\[ P(\text{Not Crash}) = 1 - 0.00003 = 0.99997 \]

5. **Final Answer:**
The probability that the aircraft will not crash is \( 0.99997 \).