Ranjani and Kanika are friends. What is the probability that both will have:
(i) different birthdays?
(ii) the same birthday? (ignoring a leap year)

Answer

To solve the problem of finding the probability that Ranjani and Kanika have different birthdays and the probability that they have the same birthday, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Total Number of Days in a Year:**
- In a normal year (ignoring leap years), there are 365 days.
- **Hint:** Remember that we are considering a standard year without February 29.

2. **Calculate the Probability of Having Different Birthdays:**
- If Ranjani has a birthday on any of the 365 days, Kanika can have her birthday on any of the remaining days.
- Therefore, if Ranjani's birthday is fixed, Kanika has 364 options left for her birthday.
- The total number of possible birthday combinations for both is \(365 \times 365\) (since each can have any of the 365 days).
- The number of favorable outcomes for different birthdays is 364 (the number of choices Kanika has after Ranjani's birthday is chosen).
- Thus, the probability \(P(\text{different birthdays})\) is given by:
\[ P(\text{different birthdays}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{364}{365} \]

3. **Calculate the Probability of Having the Same Birthday:**
- For Ranjani and Kanika to have the same birthday, there is only one favorable outcome (both having their birthday on the same day).
- The total number of possible birthday combinations remains \(365 \times 365\).
- Thus, the probability \(P(\text{same birthday})\) is given by:
\[ P(\text{same birthday}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{365} \]

### Final Answers:
- (i) The probability that Ranjani and Kanika have different birthdays is \(\frac{364}{365}\).
- (ii) The probability that Ranjani and Kanika have the same birthday is \(\frac{1}{365}\).