A machine operates only when all of its three components function. The probabilities of the failures of the first, second and third components are 0.14,0.10 and 0.05 respectively. What is the probability that the machine will fail.

Answer

To solve the problem of finding the probability that the machine will fail, we can follow these steps:

### Step 1: Define Events
Let:
- A = event that the first component fails
- B = event that the second component fails
- C = event that the third component fails

### Step 2: Identify Probabilities of Failure
From the problem, we know:
- Probability of failure of the first component, \( P(A) = 0.14 \)
- Probability of failure of the second component, \( P(B) = 0.10 \)
- Probability of failure of the third component, \( P(C) = 0.05 \)

### Step 3: Calculate Probabilities of Success
We need to find the probabilities that each component is functioning (not failing):
- Probability that the first component is functioning, \( P(A') = 1 - P(A) = 1 - 0.14 = 0.86 \)
- Probability that the second component is functioning, \( P(B') = 1 - P(B) = 1 - 0.10 = 0.90 \)
- Probability that the third component is functioning, \( P(C') = 1 - P(C) = 1 - 0.05 = 0.95 \)

### Step 4: Calculate the Probability that All Components Function
The machine will operate only if all components are functioning. Therefore, we calculate the probability that all components are functioning:
\[ P(\text{Machine works}) = P(A') \times P(B') \times P(C') = 0.86 \times 0.90 \times 0.95 \]

### Step 5: Perform the Calculation
Calculating the above expression:
\[ P(\text{Machine works}) = 0.86 \times 0.90 = 0.774 \]
\[ P(\text{Machine works}) = 0.774 \times 0.95 = 0.7353 \]

### Step 6: Calculate the Probability that the Machine Fails
The probability that the machine fails is the complement of the probability that it works:
\[ P(\text{Machine fails}) = 1 - P(\text{Machine works}) = 1 - 0.7353 = 0.2647 \]

### Final Answer
Thus, the probability that the machine will fail is:
\[ \boxed{0.2647} \]