Final answer:
To calculate the probability of 2 or fewer relief valves opening properly, we'd use the binomial probability formula for k=0, k=1, and k=2, summing the results. Specific values for the total number of valves and the individual probability of opening correctly are required for an actual calculation.
Step-by-step explanation:
To determine the probability that 2 or fewer relief valves open properly, we must know the total number of valves tested and the probability that any individual valve will open properly. Without this specific information, we cannot calculate an accurate probability. However, if we assume that the probability of a valve opening correctly is p, then the probability of exactly k valves opening in a sample of n valves can be calculated using the binomial probability formula:
P(X=k) = \(\binom{n}{k}\) * p^k * (1-p)^(n-k)
Applying this, the probability of 2 or fewer valves opening properly (i.e., P(X ≤ 2)) is the sum of the probabilities for k=0, k=1, and k=2. To find this sum, we would use the binomial formula for each value of k and add the results together:
- P(X=0) = \(\binom{n}{0}\) * p^0 * (1-p)^n
- P(X=1) = \(\binom{n}{1}\) * p^1 * (1-p)^(n-1)
- P(X=2) = \(\binom{n}{2}\) * p^2 * (1-p)^(n-2)
Keep in mind that to perform specific calculations, concrete values for n and p are required, which are not provided in the question.