(Right) On Schedule

Last night, I got an email. It wasn't an advertisement, or a large-group message. Just an email from a colleague. It was a somewhat personal email, the kind that, as the sender, you might have felt anxious about sending. You might have schedule-sent it for a few minutes after you finished writing. Maybe you'd have picked the nearest half-hour, to give yourself a moment to read it over, and to soften the blow of clicking send with some comforting ability for takesie-backsies. As the recipient, I could only wonder. And as myself, I got to wondering a little harder than I probably ought to have.

Given that I received an email at 7:30pm, what is the probability that it was schedule-sent? We have to make some assumptions.

First, assume that the probability that any given email is scheduled is 10%. Second, assume that unscheduled email send times follow a uniform distribution. We will discretize by minute. So, the probability an unscheduled email is sent at 9:00 is the same as the probability it is sent at 9:01, or 9:02, and so on. Finally, assume that, given an email is scheduled, it is scheduled on the hour (e.g. 7:00, 8:00, etc.) with 75% probability, and on the half-hour (e.g. 7:30, 8:30, etc.) with 25% probability. Thus, we neglect the possibility that someone schedules an email for any time other than on the hour or on the half-hour. We also ignore any higher likelihood that emails are scheduled for certain popular times, such as 9:00am. Our assumptions are imperfect, but they give a simple model that is, I think, believable. “All models are wrong,” goes an adage in physics, “but some models are useful.”

Bayes’s Theorem then tells us that that for given events A and B, the probability of A occurring given that B occurred is equal to the probability of B occurring given that A occurred, times the probability of A occurring, divided by the probability of B occurring. In an equation,

\[ P(\text{A} \mid \text{B}) = \frac{P(\text{B} \mid \text{A}) \times P(\text{A})}{P(\text{B})}. \]

In this scenario, let A be “the email is schedule-sent” and let B be “it is 7:30pm”. Thus Bayes’s Theorem reads: “the probability that the email is schedule-sent given that it is 7:30pm is equal to the probability that it is 7:30pm given that the email is schedule-sent times the probability that an email is schedule-sent divided by the probability that it is 7:30pm.” Since our probability distributions do not depend on the hour of the day, we can compute probabilities for a single hour of the day, and the results will extend to all hours. So, in all the following, we take as given that it is between 7:00pm and 7:59pm, inclusive. From the assumptions,

\(P(\text{it is 7:30pm, given that the email is schedule-sent}) = 0.25\)

\(P(\text{an email is schedule-sent}) = 0.1\)

\(P(\text{it is 7:30pm})\) is the trickiest. At first glance, this would simply be 1 minute out of 60, so we would naively say \(\frac{1}{60} \approx 0.0167\). But this would lead us to calculating 150% for the probability that an email received at 7:30pm was scheduled. There is a hidden given attached to this probability: “given that I received an email.” This given does not have a uniform distribution, because we bias the 7:30 time with the schedule-sent emails. So, we must appeal to the law of total probability:

\[P(\text{A}) = P(\text{A} \mid \text{B}) \times P(\text{B}) + P(\text{A} \mid \text{C}) \times P(\text{C}).\]

Here B and C are “scheduled email” and “unscheduled email,” respectively, and A is “7:30pm” So, we have

\[ \begin{aligned} P(\text{it is 7:30pm}) &= P(\text{it is 7:30pm} \mid \text{the email was scheduled})\\ &\quad \times P(\text{the email was scheduled}) \\ &\quad + P(\text{it is 7:30pm} \mid \text{the email was unscheduled}) \\ &\quad \times P(\text{the email was unscheduled}) \end{aligned} \]

which gives

\[P(\text{it is 7:30pm}) = 0.25*0.1 + (1/60)*0.9 = 0.04.\]

It makes sense that the probability that it is 7:30pm (as opposed to any other time) is slightly higher than the probability that it is any other minute, because we secretly are taking for granted that we received an email, which is overall more likely at a half-hour than any other minute (except on-the-hour).

So, finally, given that I received an email at 7:30pm, what is the probability that it was schedule-sent? \(0.25*0.1/0.04 = 62.5\%\). Seems high, but that is what the theorems tell us, given our (imperfect) assumptions. If we repeat the same calculation for an on-hour email (received at 7:00pm), we get an even more shocking \(83\%\) chance that it was scheduled.

The assumptions are certainly imperfect, but still they make for a good toy example and confirm an intuitive suspicion that an email received precisely on the hour is likely to have be scheduled.

After some thought and apprehension of my own, I replied to the email this morning. I stared out the window for a moment, weighing whether I really wanted to irrevocably share what I had written. Clenching my teeth a little, I clicked “send.” The time? Exactly 8:30am.