Final answer:
To find P(Ac and Bc), we calculate the complements of P(A) and P(B) since A and B are independent events, then multiply these complements together. However, none of the provided answer options match the calculated value of 0.56, suggesting a possible typo or error in the question.
Step-by-step explanation:
The student's question, "What is P(Ac and Bc)?" refers to the probability of the complement of event A and the complement of event B occurring together. To find this probability, we must understand the concepts of independent events and the complement rule in probability.
Given that P(A) = .2 and P(B) = .3, and knowing that A and B are independent events, we can calculate P(A AND B) first, which is the product of their individual probabilities since they are independent. Therefore, P(A AND B) = P(A)P(B) = (0.2)(0.3) = 0.06. To find the complement, which is P(Ac and Bc), we apply the complement rule:
- P(Ac) = 1 - P(A) = 1 - 0.2 = 0.8
- P(Bc) = 1 - P(B) = 1 - 0.3 = 0.7
Since A and B are independent, their complements are also independent. Hence, P(Ac and Bc) = P(Ac)P(Bc) = (0.8)(0.7) = 0.56.
Looking at the provided options, none of them match our calculation. Thus, it may either be a trick question, we may have misunderstood the provided options or additional context, or there could have been a typo or error in the options provided. In a typical scenario, we would seek clarification or suggest reviewing the question and the options provided.