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Problem 2 In a batch of microprocessors, the probability that a microprocessor ...

Sep 4, 2024

Solution by Steps

step 1

To find the probability that a lot is rejected, we need to calculate

P(X \geq 2)

where

X

follows a binomial distribution with parameters

n = 10

and

p = 10^{-3}

. The probability of rejection is given by P(X \geq 2) = 1 - P(X < 2) = 1 - (P(X = 0) + P(X = 1))

step 2

The probabilities can be calculated using the binomial formula:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

For

P(X = 0)

P(X = 0) = \binom{10}{0} (10^{-3})^0 (1 - 10^{-3})^{10} = (0.999)^{10} \approx 0.99004

For

P(X = 1)

P(X = 1) = \binom{10}{1} (10^{-3})^1 (1 - 10^{-3})^{9} = 10 \cdot (10^{-3}) \cdot (0.999)^{9} \approx 0.00990

Thus, P(X < 2) \approx 0.99004 + 0.00990 \approx 0.99994 Therefore,

P(X \geq 2) \approx 1 - 0.99994 = 0.00006

step 3

For the second part, we need to find the probability that at least 60 microprocessors are retained after 6 draws. Each draw consists of 10 microprocessors, so we have a total of 60 microprocessors. The probability of retaining a lot (i.e., having less than 2 defective microprocessors) is P(X < 2) from the first part, which is approximately

0.99994

. The probability of retaining at least 60 microprocessors after 6 draws is given by

P(Y \geq 60)

where

Y

is the number of retained lots. Since each lot is independent, we can use the binomial distribution again with

n = 6

and

p \approx 0.99994

. Thus, P(Y \geq 60) = 1 - P(Y < 60) \approx 1

[question number] Answer

Key Concept

Binomial Distribution

Explanation

The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it helps calculate the probabilities of defective microprocessors.