Among 15 metal parts produced in a machine shop, 4 are defective. If a random sample of five of these metal parts is selected, find
a) The probability that this sample will contain at least three defectives.
b) The probability that this sample will contain at most two defective.
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To solve these problems, we can use the concept of binomial distribution. Let's go step by step.
a) To find the probability that the sample will contain at least three defectives, we need to calculate the probability of getting 3, 4, or 5 defective parts in the sample.
Step 1: Calculate the probability of getting exactly 3 defectives:
We can use the formula for binomial distribution: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the total number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the combination formula.
In our case, n = 5 (sample size), k = 3, and p = 4/15 (probability of selecting a defective part).
P(X=3) = C(5, 3) * (4/15)^3 * (1 - 4/15)^(5-3) = 10 * (4/15)^3 * (11/15)^2.
Step 2: Calculate the probability of getting exactly 4 defectives:
Using the same formula, with k = 4:
P(X=4) = C(5, 4) * (4/15)^4 * (1 - 4/15)^(5-4) = 5 * (4/15)^4 * (11/15)^1.
Step 3: Calculate the probability of getting exactly 5 defectives:
Using the same formula, with k = 5:
P(X=5) = C(5, 5) * (4/15)^5 * (1 - 4/15)^(5-5) = (4/15)^5 * (11/15)^0.
Step 4: Add the probabilities from all three steps to get the probability of having at least three defectives:
P(at least 3 defectives) = P(X=3) + P(X=4) + P(X=5).
b) To find the probability that the sample will contain at most two defective parts, we need to calculate the probabilities of getting 0, 1, or 2 defective parts in the sample.
Step 1: Calculate the probability of getting exactly 0 defectives:
Using the same formula as in step 1, with k = 0:
P(X=0) = C(5, 0) * (4/15)^0 * (1 - 4/15)^(5-0) = (11/15)^5.
Step 2: Calculate the probability of getting exactly 1 defective:
Using the same formula as in step 2, with k = 1:
P(X=1) = C(5, 1) * (4/15)^1 * (1 - 4/15)^(5-1) = 5 * (4/15)^1 * (11/15)^4.
Step 3: Calculate the probability of getting exactly 2 defectives:
Using the same formula as in step 2, with k = 2:
P(X=2) = C(5, 2) * (4/15)^2 * (1 - 4/15)^(5-2) = 10 * (4/15)^2 * (11/15)^3.
Step 4: Add the probabilities from all three steps to get the probability of having at most two defectives:
P(at most 2 defectives) = P(X=0) + P(X=1) + P(X=2).