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To solve this problem, we need to find the probability that a five-day school week contains at least one snow day, given that the probability of a snow day on any given day is \( \frac{1}{4} \).

### Step-by-step solution:

1. **Define the given probabilities:**
   - The probability that a snow day *does* happen on a given day is \( P(\text{snow day}) = \frac{1}{4} \).
   - The probability that a snow day *does not* happen on a given day is \( P(\text{no snow day}) = 1 - \frac{1}{4} = \frac{3}{4} \).

2. **Find the probability of no snow day in a 5-day week:**
   We want to calculate the probability that there are no snow days in the entire five-day week. Since the events are independent, we multiply the probabilities of no snow day for each of the five days:
   \[
   P(\text{no snow days in 5 days}) = \left( \frac{3}{4} \right)^5 = \frac{3^5}{4^5} = \frac{243}{1024}.
   \]

3. **Find the probability of at least one snow day:**
   The probability of at least one snow day is the complement of the probability of no snow days. That is:
   \[
   P(\text{at least one snow day}) = 1 - P(\text{no snow days in 5 days}) = 1 - \frac{243}{1024} = \frac{1024 - 243}{1024} = \frac{781}{1024}.
   \]

So, the probability that there is at least one snow day in a five-day school week is \( \frac{781}{1024} \), or approximately 0.7627 (about 76.27%).

Would you like more details or further explanations?

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Here are five related questions to expand on this topic:
1. What is the probability of exactly one snow day in a five-day week?
2. How would the result change if the probability of a snow day were \( \frac{1}{3} \) instead of \( \frac{1}{4} \)?
3. What is the probability of having snow on all five days?
4. How would you compute the expected number of snow days in a week?
5. How does this problem relate to binomial probability distribution?

**Tip**: When dealing with "at least one" problems, it's often easier to compute the complement (no occurrences) and subtract it from 1!

In the town of Winterburg, the probability that it will snow enough to cancel school on any given day in January is 1/4. What is the probability that a given five-day school week contains at least one such snow day?

This problem explores how to find the probability that there is at least one snow day during a five-day school week, given that the probability of a snow day on any given day is 1/4. The solution involves key probability concepts, including the complement rule and independent events.

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