Suppose the probability that it rains in the next two days is one-third for tomorrow and start fraction 1 over 6 end fraction for the day after tomorrow. What is P(rain tomorrow, then rain the day after tomorrow)?
A. one-half
B. one-eighteenth
C. two-ninths
D. one-ninth
We can use the multiplication rule for independent events, which states that the probability of two independent events occurring together is the product of their individual probabilities. In this case, we can let R1 and R2 denote the events "it rains tomorrow" and "it rains the day after tomorrow", respectively. Then we have:
P(R1 and R2) = P(R1) * P(R2 | R1)
where P(R2 | R1) is the conditional probability of R2 given that R1 has occurred. To find this probability, we can use the fact that the events are independent and multiply:
P(R2 | R1) = P(R1 and R2) / P(R1) = (1/3) * (1/6) / (1/3) = 1/6
Therefore, we have:
P(R1 and R2) = P(R1) * P(R2 | R1) = (1/3) * (1/6) = 1/18
So the answer is (B) one-eighteenth.