There is 25% chance that it rains on any particular day. What is probability that there is at least one rainy day within a period of 7days?

Options

A. `1-((1)/(4))^(2)`
B. `((1)/(4))^(7)`
C. `((3)/(4))^(7)`
D. `1-((3)/(4))^(7)`

Answer

To solve the problem, we need to find the probability that there is at least one rainy day within a period of 7 days, given that there is a 25% chance of rain on any particular day.

### Step-by-Step Solution:

1. **Identify the probability of rain on a single day**:
The probability that it rains on any particular day is given as 25%, which can be expressed as a fraction:
\[ P(\text{Rain}) = \frac{25}{100} = \frac{1}{4} \]

2. **Calculate the probability of no rain on a single day**:
The probability of no rain on a particular day is the complement of the probability of rain:
\[ P(\text{No Rain}) = 1 - P(\text{Rain}) = 1 - \frac{1}{4} = \frac{3}{4} \]

3. **Calculate the probability of no rain over 7 days**:
Since the days are independent, the probability of no rain for 7 consecutive days is:
\[ P(\text{No Rain in 7 days}) = \left(P(\text{No Rain})\right)^7 = \left(\frac{3}{4}\right)^7 \]

4. **Calculate the probability of at least one rainy day in 7 days**:
The probability of having at least one rainy day in the 7-day period is the complement of having no rain at all in those 7 days:
\[ P(\text{At least 1 Rainy Day}) = 1 - P(\text{No Rain in 7 days}) = 1 - \left(\frac{3}{4}\right)^7 \]

5. **Final Calculation**:
Now, we can calculate the final probability:
\[ P(\text{At least 1 Rainy Day}) = 1 - \left(\frac{3}{4}\right)^7 \]