Final answer:
To ensure at least a 0.995 probability that one component operates correctly with each component having a 0.90 probability of operating, you would need at least 3 components.
Step-by-step explanation:
The question asks to determine the number of components needed to achieve a specific probability that at least one component functions correctly. Given that each component has a probability of 0.90 of operating correctly and the events are independent, the probability that a component fails is 0.10. To find the value of n for which there is at least a 0.995 probability that at least one component operates correctly, we use the complement rule, which states that the probability of at least one success is 1 minus the probability of all failures.
Let's denote the probability of all components failing as P(all fail). Since the components fail independently, the probability that all n components fail is (0.10)^n. We want the complement of this to be at least 0.995, so:
1 - (0.10)n ≥ 0.995
Thus, (0.10)n ≤ 0.005.
Taking the logarithm of both sides gives us:
n * log(0.10) ≤ log(0.005)
Dividing by log(0.10), which is negative, reverses the inequality:
n ≥ log(0.005) / log(0.10)
Calculating the right side gives us:
n ≥ 2 / (-1) = -2
Since n must be a whole number, we round up to get n ≥ 3. Hence, you need at least 3 components to be at least 99.5% certain that at least one operates correctly.