There is a 3/8 chance that it will rain on one day. Find the probability that, over two days, it rains on exactly one of those days

To solve this problem, we need to calculate the probability that it will rain on exactly one of two days, given that the probability of rain on any single day is \( \frac{3}{8} \).

Let's define the events:

- Let \( P(\text{Rain}) = \frac{3}{8} \) be the probability that it rains on a given day.
- Let \( P(\text{No Rain}) = 1 - P(\text{Rain}) = 1 - \frac{3}{8} = \frac{5}{8} \) be the probability that it does not rain on a given day.

We want to find the probability that it rains on exactly one of the two days. There are two possible scenarios:

1. It rains on the first day and does not rain on the second day.
2. It does not rain on the first day, but it rains on the second day.

### Calculation

1. **Probability that it rains on the first day and does not rain on the second day:**

   \[
   P(\text{Rain on Day 1 and No Rain on Day 2}) = P(\text{Rain}) \times P(\text{No Rain}) = \frac{3}{8} \times \frac{5}{8}
   \]

   \[
   = \frac{15}{64}
   \]

2. **Probability that it does not rain on the first day, but it rains on the second day:**

   \[
   P(\text{No Rain on Day 1 and Rain on Day 2}) = P(\text{No Rain}) \times P(\text{Rain}) = \frac{5}{8} \times \frac{3}{8}
   \]

   \[
   = \frac{15}{64}
   \]

### Adding the Probabilities

To find the probability that it rains on exactly one of the two days, we add the probabilities of the two scenarios:

\[
P(\text{Exactly one rainy day}) = \frac{15}{64} + \frac{15}{64} = \frac{30}{64}
\]

\[
= \frac{15}{32}
\]

### Conclusion

The probability that it rains on exactly one of the two days is \( \frac{15}{32} \).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. What is the probability that it rains on both days?
2. What is the probability that it does not rain on either of the two days?
3. If the probability of rain changes to \( \frac{1}{2} \), what would the probability be that it rains on exactly one of the two days?
4. How would the probability change if we considered three days instead of two?
5. What is the expected number of rainy days over a two-day period with the given probability?

**Tip:** When dealing with multiple independent events, consider each possible scenario separately, calculate their probabilities, and then combine them according to the problem requirements.

Learn how to calculate the probability that it rains on exactly one day out of two, given a probability of rain on a single day. This problem involves applying probability theory to independent events and includes step-by-step calculations. Suitable for Grades 9-12.

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