What is the probability of the same event happening to two (or more) different independent people within a specified time frame?
For example, if I wave my hands to my friend, there is a probability of 1/2 that he will wave back after a second. Or, he will wave back after 2 seconds with the same probability. What is the probability of two of my friends waving back at the same time at a specific instant (assuming the same probability and timing of 1-2 seconds)? How about three or more?
time p1 p2 probability
1s wave wave 1/2 -> probability of both p1 and p2 to wave back here!
2s wave wave 1/2 -> or here
I know the more people there are, the higher the probability for more of my friends to wave back to me at the same time. But how do I prove this mathematically?
Best Answer
I am not sure that I understood your question right. We can consider the following general model. Your meet $n$ friends and wave hands to them. Then each of them have will wave back, after $1$, $2$, ... or $n$ seconds, with equal probability $p=1/n$. There are $n^n$ possible waving delay sequences $(d_1,\dots, d_n)$, where $d_i$ is a waving delay for each friend, and each of these sequences has probability $n^{-n}$. Then the probability that all friends will wave back to you at the same time is $n^{1-n}$, since there are only $n$ such waving delay sequences, namely, $(1,1,\dots,1), (2,2,\dots,2),\dots, (n,n,\dots,n)$.