What are the chances she'll email me back?

I installed a nifty little gadget on my gmail client the other month, which reports different statistics about my emailing habits. It's pretty easy to install, as long as you don't mind giving this script permission to access your account. Every month, I get an automated email from myself describing the behavior of my sent and received emails.

One graph which surprised me was the distribution of "conversation" lengths in my inbox. A conversation is defined as any new email and subsequent responses. (an email with the subject "Re: Re: Re: photos" is part of the conversation "photos".) So the more responses to an email, the larger the conversation is. Here's the distribution of conversations in my inbox for July.

It's plotted on a log-log graph. and every point falls on a line, much like a power function. This suggests that there is simply a fixed probability with which someone responds to an email. In other words, the probability that I get a response to a message is independent of how long the conversation already is.

Lets do some math... p(n) is the probability of having a conversation of length n. If we humans really reply with a time-independent fixed probability, then we expect the distribution p(n) to be equal to the probability of a single message being sent n times. I call this basal response probability p1. 

Taking the log on both sides, we can bring down the exponent and see how this distribution would be linear on a log-log plot.

So what is the probability that I send/receive a single email? Lets solve for p1! Plugging in some numbers from the graph above, I find that p1 = 4.4%.

Of course, we humans like to think that we make conscious decisions, and that we answer emails based on their relative importance. But the fact that my conversation distribution consistently falls on this line means one thing: either the vast majority of my responses are canned and trivial, or worse, I answer important emails at the expense of answering less important conversations less frequently than I would otherwise. 

I would be interested in seeing how this basal response probability (4.4%) compared to other people. Some other questions to pursue: do I preferentially end conversations, or do my pen pals? Does my p1 change from month to month? Is the variation away from a linear log-log behavior a good indicator for my productivity/procrastination for that month? Is this true for texts?

Maybe the most important takeaway here is that if the person didn't respond to your message, you can expect to send your message about twenty-five more times before they'll reply once. :)