What is the overall penetration value (p(overall)) for four control devices with 90% efficiency each?

Steps to Solve

1. Identify the efficiency of each device Each control device has an efficiency of 90%. In decimal form, this is represented as: $$ p = 0.90 $$

2. Calculate the probability of failure for each device To find the probability that a device does not function (fails), we subtract the efficiency from 1: $$ q = 1 - p = 1 - 0.90 = 0.10 $$

3. Find the overall probability of failure for all devices Since the devices operate independently, we can multiply the probabilities of failure for each device. There are four devices: $$ P_{\text{failure}} = q^4 = (0.10)^4 $$

4. Calculate the overall penetration value To find the overall efficiency (or penetration value), we subtract the overall failure probability from 1: $$ P_{\text{penetration}} = 1 - P_{\text{failure}} $$

5. Perform the calculations First, calculate the probability of failure: $$ P_{\text{failure}} = (0.10)^4 = 0.0001 $$

Now, calculate the penetration value: $$ P_{\text{penetration}} = 1 - 0.0001 = 0.9999 $$