Question

a shipment of 12 microwave ovens contains 3 defective units. a restaurant buys three of these units. what is the probability of the restaurant buying at least two nondefective units?

the probability of the restaurant buying at least two nondefective units is .
(round to three decimal places as needed.)

Explanation:

Identify the parameters of the problem

Using the Hypergeometric Distribution and Combinatorics knowledge points

• Total shipment size \(N = 12\)
• Defective units \(d = 3\)
• Nondefective units \(g = N - d = 9\)
• Sample size selected \(n = 3\)
• Target variable \(X\): number of nondefective units purchased

Determine the favorable outcomes

Using the Combinatorics knowledge point

• We want the probability of buying at least two nondefective units: \(P(X \ge 2) = P(X = 2) + P(X = 3)\).
• Total ways to choose 3 units from 12:

$$ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 $$

Calculate the probability of exactly two nondefective units

Using the Combinatorics knowledge point

• Ways to choose 2 nondefective from 9 and 1 defective from 3:

$$ \binom{9}{2} \times \binom{3}{1} = \frac{9 \times 8}{2 \times 1} \times 3 = 36 \times 3 = 108 $$

• Probability \(P(X = 2)\):

$$ P(X = 2) = \frac{108}{220} $$

Calculate the probability of exactly three nondefective units

Using the Combinatorics knowledge point

• Ways to choose 3 nondefective from 9 and 0 defective from 3:

$$ \binom{9}{3} \times \binom{3}{0} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} \times 1 = 84 \times 1 = 84 $$

• Probability \(P(X = 3)\):

$$ P(X = 3) = \frac{84}{220} $$

Sum the probabilities and round to three decimal places

Using the Hypergeometric Distribution knowledge point

• Total favorable outcomes: \(108 + 84 = 192\)
• Total probability:

$$ P(X \ge 2) = \frac{192}{220} = \frac{48}{55} \approx 0.872727... $$

• Rounded to three decimal places, the probability is \(0.873\).

Answer:

The probability of the restaurant buying at least two nondefective units is <blank>0.873</blank>.
(Round to three decimal places as needed.)