# Probability That No One Has A Specific Birthday and At Least One Person Does

The question is as follows:

What is the probability that $$1017$$ persons in a certain building will say that they were not born on July $$15$$? What is the probability that someone (meaning at least one person) will say July $$15$$?

My approach...

The probability that one person does not have their birthday fall on July $$15$$ is $$\frac{364}{365}$$, assuming that the year is not a leap year. For all 1017 persons, I thought of raising $$\frac{364}{365}$$ to the $$1017$$th power: $$(\frac{364}{365})^{1017}$$.

The second portion of the question made me think that if there is someone who has a birthday on July $$15$$, then the remaining 1016 will have the probability of $$\frac{364}{365}$$ and the one person will have the probability of $$\frac{1}{365}$$. However, I am unsure because the equation $$(\frac{364}{365})^{1016}(\frac{1}{365})$$ will represent the probability for exactly one person to have their birthday on that date, not at least one person.

Any help will be greatly appreciated.

• Are you assuming that there are 1017 people in the building? And that all are not born on July 15. Or are you saying there are $n$ number of people in the building, 1017 weren't born on July 15 and $n -1017$ were? – fleablood Oct 6 '18 at 19:29
• "assuming that the year is not a leap year." I'm going to nitpick that whether this year is a leap year has no bearing whatsoever when someones birthday is. My birthday is the same this year as it was in 2016. If I had been born on Feb 29 on a leap year that would still be my birthday no matter what year you asked me..... – fleablood Oct 6 '18 at 19:33
• @fleablood If it was a leap year, I would have 366 days in total - that's all I meant - and the problem mentioned to assume that it was not a leap year if anyone was confused. – geo_freak Oct 6 '18 at 19:36
• ... that said, we can say "for simplicity we can ignore leap days" or we can say that ans 1 in $4*365+1$ the probability of not July 15 is $\frac {4*364 + 1}{4*365 + 1}$ But that is close enough to $\frac {364}{365}$ to not be fussy. After all, Not every day is equally likely (just look at any census) and the assumption that they are equal is no more inaccurate than the rounding down. – fleablood Oct 6 '18 at 19:38
• "If it was a leap year, I would have 366 days in total" and ... so?.... How would that affect the birthdays of the people in the building? Or they all under 1 year old? We don't care about whether this year is a leap year. We care about whether the year that the tenant was born was a leap year. The problem says, don't worry about it, which is fair. But assuming this year isn't a leap year isn't the way to avoid it. The way to avoid it is by assuming no tenant was born on Feb 29. Or to assume, for the sake of this excercise, Feb 29 does not exist in this universe. – fleablood Oct 6 '18 at 19:43

Your solution to the first part is correct. As for the second part just note that "at least one person was born that day" is the complement of "none of them were born on that day". So the answer is just $$1-(\frac{364}{365})^{1017}$$.