In a group of 3 persons, if p is the probability that at least two of them have the same birthdays, then `(365)^p` - 1,085 is equal to _____

Answer

To solve the problem, we need to find the probability \( p \) that at least two out of three persons have the same birthday, and then calculate \( 365^p - 1085 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the probability \( p \) that at least two out of three persons have the same birthday. This can be calculated using the complement rule.

2. **Complement of the Event**:
The complement of "at least two persons have the same birthday" is "all three persons have different birthdays". Therefore, we can express \( p \) as:
\[ p = 1 - P(\text{all different birthdays}) \]

3. **Calculating \( P(\text{all different birthdays}) \)**:
For the first person, there are 365 choices for their birthday. For the second person, to have a different birthday, there are 364 choices. For the third person, there are 363 choices. Thus, the total number of ways for three persons to have different birthdays is:
\[ 365 \times 364 \times 363 \]
The total number of possible birthday combinations for three persons is:
\[ 365^3 \]
Therefore, the probability that all three have different birthdays is:
\[ P(\text{all different}) = \frac{365 \times 364 \times 363}{365^3} \]

4. **Calculating \( P(\text{all different}) \)**:
Simplifying this expression:
\[ P(\text{all different}) = \frac{364}{365} \times \frac{363}{365} \]
Calculating this gives:
\[ P(\text{all different}) \approx 0.9918 \]

5. **Finding \( p \)**:
Now, substituting back to find \( p \):
\[ p = 1 - P(\text{all different}) = 1 - 0.9918 \approx 0.0082 \]

6. **Calculating \( 365^p - 1085 \)**:
Now we need to calculate \( 365^p \):
\[ 365^{0.0082} \approx 1.028 \]
Finally, we subtract 1085 from this value:
\[ 365^{0.0082} - 1085 \approx 1.028 - 1085 \approx -1083.972 \]

### Final Answer:
The value of \( 365^p - 1085 \) is approximately \( -1083.972 \).