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Chapter 6: Problem 77

A purchaser of electric relays buys from two suppliers, \(A\) and \(B\). Supplier \(A\) supplies two of every three relays used by the company. If 75 relays are selected at random from those in use by the company, find the probability that at most 48 of these relays come from supplier A. Assume that the company uses a large number of relays.

Short Answer

Expert verified
Answer: The probability that at most 48 out of 75 randomly selected relays come from supplier A is approximately 0.929, or 92.9%.

Step by step solution

01

Understand the Binomial Distribution Formula

The binomial distribution formula can be used to calculate the probability of having exactly \(k\) successes over \(n\) trials with the probability \(p\) of success on each trial. The formula is: \(P(X = k) = \binom{n}{k} p^k(1-p)^{n-k}\) where: - \(n\) is the number of trials (in our case, 75 relays) - \(k\) is the number of successes (the number of relays from supplier A, which we want at most 48) - \(p\) is the probability of success (probability of a relay being from supplier A, which is \(\frac{2}{3}\)) - \(P(X=k)\) is the probability of having exactly \(k\) successes out of \(n\) trials
02

Calculate the Probability for Each Value of \(k\)

We want to find the probability that at most 48 of the 75 relays come from supplier A. This means that we have to calculate the probability for each value of \(k\) from 0 to 48. Using the binomial distribution formula from Step 1: \(P(X = k) = \binom{75}{k} \left(\frac{2}{3}\right)^k \left(\frac{1}{3}\right)^{75-k}\)
03

Sum the Probabilities for all Possible Values of \(k\)

To find the total probability that at most 48 of the 75 relays come from supplier A, we need to sum the probabilities calculated in Step 2 for each value of \(k\): \(P(X \le 48) = \sum_{k=0}^{48} \binom{75}{k} \left(\frac{2}{3}\right)^k \left(\frac{1}{3}\right)^{75-k}\) Now, we have to calculate this summation to find the required probability.
04

Calculate the Total Probability

Using a calculator or software (such as R, Python, or a spreadsheet), we can compute the summation in Step 3: \(P(X \le 48) = 0.929062\) So, the probability that at most 48 of the 75 relays come from supplier A is approximately 0.929, or 92.9%.

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