A manufacturing process has a 90% yield, meaning that 90% of the products are acceptable and the remainder are defective. If 4 of the products are randomly selected, find the probability that all of them are acceptable.
The probability of all events occurring is found by multiplying the individual probabilities.
(.9)(.9)(.9)(.9) = ?
To find the probability that all four products are acceptable, we need to use the concept of probability and the given yield (success) rate.
Step 1: Understand the problem
We are given that the manufacturing process has a 90% yield, which means that 90% of the products are acceptable. This implies that the probability of selecting an acceptable product is 0.9 (90%).
Step 2: Define the event
Let's define the event A as "selecting an acceptable product."
Step 3: Determine the probability of the event happening once
We know that the probability of selecting an acceptable product is 0.9. Therefore, the probability of selecting an acceptable product once is 0.9.
Step 4: Calculate the probability of all four products being acceptable
Since the products are selected randomly, we can assume that the events of selecting each of the four products are independent. This means that the probability of all four products being acceptable is the product of the probabilities of each individual event.
P(all four products are acceptable) = P(A) * P(A) * P(A) * P(A)
Since each event has the same probability, and all four events are independent, we can calculate:
P(all four products are acceptable) = 0.9 * 0.9 * 0.9 * 0.9
Step 5: Calculate the probability of all four products being acceptable
P(all four products are acceptable) = 0.9^4
= 0.6561
Therefore, the probability that all four products selected are acceptable is 0.6561 or 65.61%.