A manufacturer of processing chips knows that 2/%2%2, percent of its chips are defective in some way. Suppose an inspector randomly selects 444 chips for an inspection. Assuming the chips are independent, what is the probability that at least one of the selected chips is defective? Lets break this problem up into smaller pieces to understand the strategy behind solving it.

The data in the question seems a bit erroneous. I am writing the correct question below:

A manufacturer of processing chips knows that 2%, percent of its chips are defective in some way. Suppose an inspector randomly selects 4 chips for an inspection. Assuming the chips are independent, what is the probability that at least one of the selected chips is defective? Lets break this problem up into smaller pieces to understand the strategy behind solving it.

Answer:

The probability that at least one of the selected chips is defective is 0.0776.

Step-by-step explanation:

The question states that the probability of defective chips is 2% i. e. 0.02. Let p denote the probability of selecting a defective chip so, p = 0.02

An inspector selects 4 chips, which means n=4 and we need to compute the probability that at least one of the selected chips is defective. Let X be the number of defective chips selected. We need to compute P (X≥1) which means either 1, 2, 3 or 4 chips can be defective.

We will use the binomial distribution formula to solve this problem. The formula is:

P (X=x) = ⁿCₓ pˣ qⁿ⁻ˣ

where n = total no. of trials

p = probability of success

x = no. of successful trials

q = probability of failure = 1-p

we have n=4, p=0.02 and q=1-0.02=0.98.

We need to compute P (X≥1) which is equal to:

P (X≥1) = P (X=1) + P (X=2) + P (X=3) + P (X=4)

A shorter method to do this is to use the total probability theorem:

P (X≥1) = 1 - P (X<1)

= 1 - P (X=0)

= 1 - ⁴C₀ (0.02) ⁰ (0.98) ⁴⁻⁰

= 1 - (0.98) ⁴

= 1 - 0.9224

P (X≥1) = 0.0776