Question

In a set of 9 electrical components, one is known to be malfunctioning. If 4 components are chosen at random, determine the probability that at least one of them is faulty. Conduct a step-by-step analysis of the possible scenarios, considering the different combinations of selecting functional and malfunctioning components. Express the probability as a decimal and round to the appropriate number of decimal places.

Answer 1

The probability that at least one of four randomly chosen electrical components from a set of nine (with one known malfunctioning) is faulty is approximately 0.4444.

Step-by-step explanation:

To find the probability that at least one of the four components chosen is faulty in a set of 9 components (with one known faulty component), we can calculate the complement of the probability that all chosen components are functional.

Step-by-step Analysis

1. Calculate the total number of ways to choose 4 components out of 9, which is the combination 9C4.
2. Determine the number of ways to choose 4 functional components out of the 8 that are functional, which is the combination 8C4.
3. Find the probability that all 4 chosen components are functional by dividing the number of functional combinations by the total combinations.
4. Calculate the probability that at least one is faulty by subtracting the probability of all functional components from 1 (the complement rule).

Calculations are as follows:

1. 9C4 = 9! / (4! * (9-4)!) = 126
2. 8C4 = 8! / (4! * (8-4)!) = 70
3. Probability of all functional = 70 / 126 ≈ 0.5556
4. Probability of at least one faulty = 1 - 0.5556 ≈ 0.4444

Therefore, the probability that at least one of the components selected is faulty is 0.4444, rounded to four decimal places.