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A factory produces bolts, and 12% are defective. A quality inspector randomly selects 15 bolts from a large...

8. A factory produces bolts, and 12% are defective. A quality inspector randomly selects 15 bolts from a large shipment. Let X be the number of defective bolts selected.
(a) Identify n and p.
(b) Find the probability that exactly 2 of the selected bolts are defective.
(c) Find the probability that at most 3 of the selected bolts are defective.
(d) Find the probability that at least 1 of the selected bolts is defective.

Answer

  1. n = 15, p = 0.12

  2. Use the binomial probability formula:
    P(X = 2) = C(15, 2) * (0.12)^2 * (0.88)^13 ≈ 0.2503

  3. P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
    Calculate each using the binomial formula and sum them:
    P(X ≤ 3) ≈ 0.7821

  4. P(X ≥ 1) = 1 - P(X = 0)
    P(X = 0) = C(15, 0) * (0.12)^0 * (0.88)^15 ≈ 0.1477
    P(X ≥ 1) ≈ 0.8523

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