Let’s solve the problem step by step using the probabilities provided:

Step 1: Define probabilities

1. If it snows on a day, the probability it will snow the next day = $\frac{7}{10}$.
2. If it does not snow on a day, the probability it will snow the next day = $\frac{2}{10}$.
3. The complement probabilities:
   • Probability it does not snow the next day given it snows today = $1 - \frac{7}{10} = \frac{3}{10}$.
   • Probability it does not snow the next day given it does not snow today = $1 - \frac{2}{10} = \frac{8}{10}$.

Step 2: Tree diagram probabilities

Today it is snowing. From there, the possible outcomes over the next two days are:

First branch (Day 1):

• Snow (S1): Probability = $\frac{7}{10}$.
• No snow (N1): Probability = $\frac{3}{10}$.

Second branch (Day 2), following from Day 1 outcomes:

1. If Day 1 = Snow (S1):

   • Snow again on Day 2 (S2): $\frac{7}{10}$.
   • No snow on Day 2 (N2): $\frac{3}{10}$.

2. If Day 1 = No snow (N1):

   • Snow on Day 2 (S2): $\frac{2}{10}$.
   • No snow on Day 2 (N2): $\frac{8}{10}$.

Step 3: Find the probability of snowing on exactly one of the next two days

To have snow on exactly one day, the following scenarios are valid:

1. Snow on Day 1 (S1) but not Day 2 (N2).
2. No snow on Day 1 (N1) but snow on Day 2 (S2).

Calculate probabilities for these cases:

1. Case 1: Snow on Day 1 but not Day 2 $P(S1 \cap N2) = P(S1) \cdot P(N2 | S1) = \frac{7}{10} \cdot \frac{3}{10} = \frac{21}{100}$

2. Case 2: No snow on Day 1 but snow on Day 2 $P(N1 \cap S2) = P(N1) \cdot P(S2 | N1) = \frac{3}{10} \cdot \frac{2}{10} = \frac{6}{100}$

Step 4: Add the probabilities

The total probability of snowing on exactly one of the next two days is: $P(\text{exactly one}) = P(S1 \cap N2) + P(N1 \cap S2) = \frac{21}{100} + \frac{6}{100} = \frac{27}{100}$

Final Answer:

The probability that it will snow on exactly one of the next two days is: $\boxed{\frac{27}{100}}$

Let me know if you want a deeper explanation or have questions! Here are some related questions to consider:

1. How can you extend this problem to three days?
2. What is the probability it snows on at least one of the next two days?
3. How would the tree diagram change if probabilities were not fixed (e.g., dependent on weather conditions)?
4. Can we calculate the expected number of snow days in two days?
5. How does this problem relate to Markov chains?

Tip: Tree diagrams are helpful visual tools for organizing multistep probability problems systematically!