A man invited five friends. He was born in April as also all the invited friends. What is the probability that none of the friends was born on the same day of the month as the host?
The way I approached it was $\frac{(30\times 29^5)}{(30^6)}$. However, there is yet another equally convincing way i.e. Probability that a friend's birthday is on the same day as the host is $\frac{1}{30}$. So if this goes for all friends then we have $\big(\frac{1}{30}\big)^5$. And we want the negation of it so $1-\big(\frac{1}{30}\big)^5$.
Which one is correct?
Best Answer
Expanding on the comments...
There are a total of $30$ days, where each day is independent of the other. The probability that the host is born on the $n_{th}$ day is $Pr(X = n) = \frac{1}{30}$, if the host is equally likely to be born on any day of this month. The probability that each friend is born on the $n_{th}$ day is also $\frac{1}{30}$ and similarly not on the $n_{th}$ day, is $\frac{29}{30}$. So it would be $(\frac{29}{30})^5$. As for $1 - (\frac{1}{30})^5$, this just means your including everything else except the all of the his friends being born on the same day. Note that set does not just include the events: all the friends being born on the same day or none of them being born on the same day.
Alternatively, the set is binomially distributed. What if $4$ friends are born on the same day and $1$ is not or $2$ friends are born on the same day and $3$ are not?
In mathematical notation your stating that, $$Pr(X = \text{None of his friends}) = 1 - Pr(X = \text{All of his friends})$$ $$Pr(X = 0) = 1 - Pr(X = 5)$$ This is incorrect as $$Pr(X = 5) = 1 - Pr(X \leq 4), \text{and }$$ $$Pr(X = 0) = 1 - Pr(X \ge 1), \text{and }X\sim Bi(n, p)$$