The probability that a bulb produced in a factory will fuse after 150 days of use in 0.05.
Find the probability that out of 5 such bulbs:
(iv) At least one will fuse after 150 days of use.

Answer

To find the probability that at least one out of five bulbs will fuse after 150 days of use, we can follow these steps:

### Step 1: Identify the probabilities
Let \( P \) be the probability that a bulb will fuse after 150 days, which is given as:
\[ P = 0.05 \]
Then, the probability that a bulb will not fuse is:
\[ Q = 1 - P = 1 - 0.05 = 0.95 \]

### Step 2: Find the probability that none of the bulbs fuse
To find the probability that none of the 5 bulbs fuse, we can use the formula for the probability of none occurring in a binomial distribution:
\[ P(X = 0) = Q^n \]
where \( n = 5 \) (the number of bulbs). Thus,
\[ P(X = 0) = 0.95^5 \]

Calculating this:
\[ P(X = 0) = 0.95^5 = 0.77378 \] (approximately)

### Step 3: Find the probability that at least one bulb fuses
The probability that at least one bulb fuses is the complement of the probability that none fuse:
\[ P(X \geq 1) = 1 - P(X = 0) \]
Substituting the value we found:
\[ P(X \geq 1) = 1 - 0.77378 \]
\[ P(X \geq 1) = 0.22622 \] (approximately)

### Final Answer
Thus, the probability that at least one out of five bulbs will fuse after 150 days of use is approximately:
\[ \boxed{0.22622} \]