Final answer:
The probability that at least one of the first 30 sets of headphones produced by Soundtronics is defective is approximately 11%.
Step-by-step explanation:
We can find the probability that at least one set of headphones is defective out of the first 30 sets by first calculating the probability that none are defective and then subtracting that probability from 1 (which represents certainty or 100%). The probability that a set of headphones is not defective is 1 minus the probability that it is defective, which is 1 - 0.0039.
To find the probability that all 30 sets of headphones are not defective, we raise this probability to the power of 30:
P(all 30 are not defective) = (1 - 0.0039)30
Calculating this gives us a probability that not one of the 30 headphones is defective. We subtract this result from 1 to get the probability that at least one is defective:
P(at least one is defective) = 1 - P(all 30 are not defective)
Carrying out this calculation:
P(at least one is defective) = 1 - (1 - 0.0039)30
Using a calculator, we determine that P(at least one is defective) is approximately 0.1135, or 11.35%, which we can round to an approximate probability of 11% (Option A).