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Chapter 23: Problem 2

The probability that a machine has a lifespan of more than 7 years is \(0.85\). Twelve machines are chosen at random. Calculate the probability that (a) 10 have a lifespan of more than 7 years (b) 11 have a lifespan of more than 7 years (c) 10 or more have a lifespan of more than 7 years.

Short Answer

Expert verified
Answer: The probabilities are (a) approximately 0.2301, (b) approximately 0.0498, and (c) approximately 0.2799.

Step by step solution

01

(a) Probability of 10 machines having a lifespan of more than 7 years

Using the binomial probability formula, we can calculate the probability of 10 machines having a lifespan of more than 7 years as: \(P(X = 10) = \binom{12}{10} (0.85)^{10}(0.15)^{2}\) \(P(X = 10) = 66 (0.85)^{10} (0.15)^{2}\) \(P(X = 10) \approx 0.2301\)
02

(b) Probability of 11 machines having a lifespan of more than 7 years

Using the binomial probability formula, we can calculate the probability of 11 machines having a lifespan of more than 7 years as: \(P(X = 11) = \binom{12}{11}(0.85)^{11}(0.15)^{1}\) \(P(X = 11) = 12(0.85)^{11}(0.15)\) \(P(X = 11) \approx 0.0498\)
03

(c) Probability of 10 or more machines having a lifespan of more than 7 years

To find the probability of 10 or more machines having a lifespan of more than 7 years, we can add the probabilities calculated previously for 10 and 11 machines together: \(P(X \geq 10) = P(X = 10) + P(X = 11) \approx 0.2301 + 0.0498 \approx 0.2799\) So, the probabilities are: (a) 10 have a lifespan of more than 7 years: approximately 0.2301. (b) 11 have a lifespan of more than 7 years: approximately 0.0498. (c) 10 or more have a lifespan of more than 7 years: approximately 0.2799.

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