Final answer:
To find the probability that 2 or less relief valves open properly, we use the binomial probability formula. The probability is approximately 0.2773.
Step-by-step explanation:
To find the probability that 2 or less relief valves open properly, we can use the binomial probability formula. Let's denote the probability of each valve opening properly as p = 0.99. The probability of exactly k valves opening properly out of n valves is given by the formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k).
In this case, n (the number of valves) is 4. So to find the probability that 2 or less valves open properly, we can calculate:
P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2) = [(4 choose 0) * 0.99^0 * 0.01^4] + [(4 choose 1) * 0.99^1 * 0.01^3] + [(4 choose 2) * 0.99^2 * 0.01^2]
Simplifying this expression gives:
P(X ≤ 2) = 0.01^4 + (4 * 0.99 * 0.01^4) + [(4 choose 2) * 0.99^2 * 0.01^2] = 0.0001 + 0.0396 + 0.2376 = 0.2773
Therefore, the probability that 2 or less relief valves open properly is approximately 0.2773.