4.104 A fresh fruit juice manufacturer finds a new ingredient that allows the juice to stay fresh for a longer duration than before. The old ingredient shows that 10% of the total quantity of juice produced will stay fresh for less than a month. The expectation is that the new ingredient will reduce the percentage to 5% or less. This will help the manufacturer to increase profits by €80,000 per year. At the end of the first cycle of one month’s production, the manufacturer wants to determine if the new ingredient has been successful. The following questions are an important part of the research design.

A total of 25 new units of juice produced were randomly selected.

a. If the new batch performs the same as the past batches, what is the probability that 20 units or more successfully increases the duration of freshness to a month or more?

b. What is the probability that 23 units or more will successfully increase the duration of freshness to a month or more?

c. If the new ingredient actually increased the probability of success to 0.95 for each production, what is the probability that 20 units or more will successfully increase the duration of freshness to a month or more?

d. Given the expected improvement, what is the probability that 23 units or more will successfully increase the duration of freshness to a month or more?