To solve this problem, we can begin by constructing a tree diagram based on the given probabilities:

1. Rain on a given day (today):

   • The probability it rains the next day is $\frac{7}{10}$.
   • The probability it does not rain the next day is $\frac{3}{10}$.

2. No rain on a given day:

   • The probability it rains the following day is $\frac{4}{10}$.
   • The probability it does not rain is $\frac{6}{10}$.

Step 1: Draw the Tree Diagram

We start from today, where it is raining:

• Day 1 (next day):
  • If it rains, with probability $\frac{7}{10}$:
    • Day 2 (the following day):
      • It rains again with probability $\frac{7}{10}$.
      • It does not rain with probability $\frac{3}{10}$.
  • If it does not rain, with probability $\frac{3}{10}$:
    • Day 2:
      • It rains with probability $\frac{4}{10}$.
      • It does not rain with probability $\frac{6}{10}$.

Step 2: Calculate Probabilities for Each Path

We want to calculate the probability of exactly one of the next two days being rainy:

1. Path: Rain on Day 1, No Rain on Day 2:

   • Probability: $\frac{7}{10} \times \frac{3}{10} = \frac{21}{100}$.

2. Path: No Rain on Day 1, Rain on Day 2:

   • Probability: $\frac{3}{10} \times \frac{4}{10} = \frac{12}{100}$.

Step 3: Add the Probabilities

Since these are mutually exclusive events (either the rain happens on Day 1 or Day 2, but not both), we add the probabilities:

$\frac{21}{100} + \frac{12}{100} = \frac{33}{100}.$

Final Answer:

The probability that it will rain on exactly one of the next two days is $\frac{33}{100}$.

Would you like further clarification, or should I break down any part of this solution in more detail?

Here are 5 related questions for further practice:

1. What is the probability that it will rain on both of the next two days?
2. What is the probability that it will not rain on either of the next two days?
3. How would the probabilities change if the chances of rain on a rainy day were $\frac{8}{10}$?
4. What is the probability that it rains on at least one of the next two days?
5. How