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

This problem can be solved using the concept of conditional probability and Markov chains. Here are the steps:

1. Define the events: Let's denote S1 as the event that 1st January is sunny, S3 as the event that 3rd January is sunny, and R1 as the event that 1st January is rainy.

2. We are asked to find P(R1|S3), the probability that 1st January is rainy given that 3rd January is sunny.

3. From the problem, we know that P(S1) = 0.8, so P(R1) = 1 - P(S1) = 0.2.

4. We also know that the probability that a sunny day is followed by a sunny day is 0.75, so the probability that a sunny day is followed by a rainy day is 1 - 0.75 = 0.25. Similarly, the probability that a rainy day is followed by a sunny day is 1 - 0.7 = 0.3.

5. Now, we can find P(S3) as follows: P(S3) = P(S1) * P(S2|S1) * P(S3|S2) + P(R1) * P(S2|R1) * P(S3|S2) = 0.8 * 0.75 * 0.75 + 0.2 * 0.3 * 0.3 = 0.45 + 0.018 = 0.468.

6. We can also find P(S3|R1) as follows: P(S3|R1) = P(S2|R1) * P(S3|S2) = 0.3 * 0.75 = 0.225.

7. Finally, we can find P(R1|S3) using Bayes' theorem: P(R1|S3) = P(S3|R1) * P(R1) / P(S3) = 0.225 * 0.2 / 0.468 = 0.096.

So, the probability that 1st January is rainy given that 3rd January is sunny is approximately 0.096 or 9.6%.

To solve this problem, we need to use the concept of conditional probability and the total probability theorem.

Let's denote: S1: the event that 1st January is sunny S2: the event that 2nd January is sunny S3: the event that 3rd January is sunny R1: the event that 1st January is rainy R2: the event that 2nd January is rainy R3: the event that 3rd 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(S1) = 0.8 (the probability that 1st January is sunny) P(S2|S1) = 0.75 (the probability that a sunny day is followed by a sunny day) P(R2|R1) = 0.7 (the probability that a rainy day is followed by a rainy day) P(S3|S2) = 0.75 (the probability that a sunny day is followed by a sunny day) P(S3|R2) = 0.3 (the probability that a rainy day is followed by a sunny day)

We can find P(S3) using the total probability theorem: P(S3) = P(S3|S2)P(S2) + P(S3|R2)P(R2) = P(S3|S2)P(S2|S1)P(S1) + P(S3|R2)P(R2|R1)P(R1) = 0.75 * 0.75 * 0.8 + 0.3 * 0.7 * (1-0.8) = 0.3375

Then we can find P(R1|S3) using Bayes' theorem: P(R1|S3) = P(S3|R1)P(R1) / P(S3) = P(S3|R2)P(R2|R1)P(R1) / P(S3) = 0.3 * 0.7 * (1-0.8) / 0.3375 = 0.0525

So, the probability that 1st January is rainy given that 3rd January is sunny is 0.0525.

To solve this problem, we need to use the concept of conditional probability and the total probability theorem.

Let's denote: S1: the event that 1st January is sunny S2: the event that 2nd January is sunny S3: the event that 3rd January is sunny R1: the event that 1st January is rainy R2: the event that 2nd January is rainy R3: the event that 3rd 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(S1) = 0.8 (the probability that 1st January is sunny) P(S2|S1) = 0.75 (the probability that a sunny day is followed by a sunny day) P(R2|R1) = 0.7 (the probability that a rainy day is followed by a rainy day) P(S3|S2) = 0.75 (the probability that a sunny day is followed by a sunny day) P(S3|R2) = 0.3 (the probability that a rainy day is followed by a sunny day)

We also know that P(R1) = 1 - P(S1) = 0.2 (the probability that 1st January is rainy)

Using the total probability theorem, we can express P(S3) as: P(S3) = P(S3|S2)P(S2) + P(S3|R2)P(R2)

We don't know P(S2) and P(R2) directly, but we can express them in terms of the events of the first day: P(S2) = P(S2|S1)P(S1) + P(S2|R1)P(R1) P(R2) = P(R2|S1)P(S1) + P(R2|R1)P(R1)

Substituting these into the equation for P(S3), we get: P(S3) = P(S3|S2)[P(S2|S1)P(S1) + P(S2|R1)P(R1)] + P(S3|R2)[P(R2|S1)P(S1) + P(R2|R1)P(R1)]

Finally, we can find P(R1|S3) using Bayes' theorem: P(R1|S3) = P(S3|R1)P(R1) / P(S3)

Substituting the known values and solving the equations will give the desired probability.