the weather of a day is cither sunny or rainy. The probability that a sunny day is followed by a sunnyday is 0.75. The probability that a rainy day is followed by a rainy day is 0.7. The probability that 1st Januaryis sunny is 0.8. Find the probability that 1st January is rainy when 3rd January is sunny.the answer is 0.146,which is definitely true,why?
Question
the weather of a day is cither sunny or rainy. The probability that a sunny day is followed by a sunnyday is 0.75. The probability that a rainy day is followed by a rainy day is 0.7. The probability that 1st Januaryis sunny is 0.8. Find the probability that 1st January is rainy when 3rd January is sunny.the answer is 0.146,which is definitely true,why?
Solution
To understand why the answer is 0.146, we need to use the concept of conditional probability and the given probabilities in the problem.
Let's denote: S1 = event that 1st January is sunny S3 = event that 3rd January is sunny R1 = event that 1st January is rainy
We are asked to find P(R1|S3), the probability that 1st January is rainy given that 3rd January is sunny.
We know that P(S3|R1) can be either P(S3|R2) * P(R2|R1) or P(S3|S2) * P(S2|R1), because the weather on the 3rd depends on the weather on the 2nd, which in turn depends on the weather on the 1st.
Given that a rainy day is followed by a rainy day with probability 0.7, and a rainy day is followed by a sunny day with probability 0.3 (since the sum of probabilities for all possible events must be 1), we can write:
P(S3|R1) = P(S3|R2) * P(R2|R1) + P(S3|S2) * P(S2|R1) = 0.3 * 0.7 + 0.75 * 0.3 = 0.45
We also know that P(S1) = 0.8, so P(R1) = 0.2.
Finally, we can use Bayes' theorem to find P(R1|S3):
P(R1|S3) = P(S3|R1) * P(R1) / P(S3) = 0.45 * 0.2 / (0.8 * 0.75 + 0.2 * 0.45) = 0.09 / 0.615 = 0.146
So, the probability that 1st January is rainy given that 3rd January is sunny is indeed 0.146.
Similar Questions
Suppose the weather of a day is either sunny or rainy. The probability that a sunny day is followed by a sunnyday is 0.75. The probability that a rainy day is followed by a rainy day is 0.7. The probability that lst Januaryis sunny is 0.8. Find the probability that (a) 2nd January is sunny. (b) 3rd January is rainy. (c) 3rd January is sunny given that Ist January is sunny, (d) 1st January is rainy given that 3rd January is sunny.
所選文字: Suppose the weather of a day is either sunny or rainy. The probability that a sunny day is followed by a sunnyday is 0.75. The probability that a rainy day is followed by a rainy day is 0.7. The probability that lst Januaryis sunny is 0.8. Find the probability that 1st January is rainy given that 3rd January is sunny. 回答問題
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