Consider a group of n people. Assume 365 days in a year and that birthdays are independent and uniformly distributed.
What is the probably that n people have no birthdays for X consecutive days?
For example, what is the probability that 31 people have no birthdays for 31 consecutive days?
Posting only because I think that the other posted answer is wrong. For $1$ specific person, the event of not having a birthday on January 1 (for example) is not independent of that person not having a birthday on January 2.
For one specific person, the probability that the person will not have a birthday in the next $X$ days, including today is
$$T = \frac{365 - X}{365}.$$
Therefore, the probability that none of $n$ people have a birthday in the next $X$ days (including today) is
$$T^n.$$