To solve this problem, we need to determine the probability that all nine flashlights selected are defective, given that the probability of a flashlight being defective is 35%, or 0.35.

We are looking for the probability that all nine flashlights are defective. Since each flashlight's defect status is independent of the others, we can multiply the individual probabilities of a single flashlight being defective:

$P(\text{all nine are defective}) = (0.35)^9$

Let's calculate this value.

$P = 0.35^9$

I'll compute the exact value for you.The probability that all nine flashlights are defective is approximately $0.00007882$, or 0.0079% when rounded to four decimal places.

Do you have any questions or would you like more details on this solution?

Here are 5 related questions for further exploration:

1. What is the probability that none of the flashlights are defective?
2. What is the probability that exactly one of the flashlights is defective?
3. What is the probability that at least one flashlight is defective?
4. How would the probability change if the defect rate was 50% instead of 35%?
5. What is the expected number of defective flashlights in a sample of nine?

Tip: When dealing with independent events, the total probability is the product of individual probabilities.