A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 55 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a \( 3 \% \) rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected? The probability that this whole shipment will be accepted is 0.5058 . (Round to four decimal places as needed.) The company will accept \( \square \% \) of the shipments and will reject \( \square \% \) of the shipments, so (Round to two decimal places as needed.)

To find the probability that the whole shipment will be accepted, we need to calculate the probability of accepting the shipment when there are no defects and when there is one defect. Given: - Total number of tablets in the shipment = 3000 - Rate of defects = 3% - Acceptance sampling plan: Accept the whole batch if there is only one or none that doesn't meet the required specifications. Let's calculate the probability of accepting the shipment when there are no defects and when there is one defect. 1. Probability of no defects: - Number of tablets without defects = 3000 * (1 - 0.03) = 3000 * 0.97 = 2910 - Probability of no defects = \( \frac{2910}{3000} \) 2. Probability of one defect: - Number of tablets with one defect = 3000 * 0.03 = 90 - Probability of one defect = \( \frac{90}{3000} \) Now, we need to calculate the probability that the whole shipment will be accepted. This can be done by adding the probabilities of no defects and one defect. Probability of accepting the shipment = Probability of no defects + Probability of one defect Let's calculate this probability. Calculate the value by following steps: - step0: Calculate: \(\left(\frac{2910}{3000}\right)+\left(\frac{90}{3000}\right)\) - step1: Reduce the fraction: \(\frac{97}{100}+\left(\frac{90}{3000}\right)\) - step2: Reduce the fraction: \(\frac{97}{100}+\frac{3}{100}\) - step3: Transform the expression: \(\frac{97+3}{100}\) - step4: Add the numbers: \(\frac{100}{100}\) - step5: Calculate: \(1\) The probability that the whole shipment will be accepted is 1, which means that almost all such shipments will be accepted. Therefore, the company will accept \( 100\% \) of the shipments and will reject \( 0\% \) of the shipments.