# The Case of M and W: How To Model This?

A man (M) and a woman (W) meet each other at an online dating website. They begin to date. After a year, M remembers that his account at the dating website is still active, and wants to delete it. So he logs in for a brief period, say 5 minutes. For the sake of this question, let us assume that the website does not allow him to permanently delete his account.

In a few minutes M receives a call from W!

W: What were you doing at that dating website?

M: I was trying to delete my account.

(Important: The only way W could have known that is that she must be logged in during that time also.)

M (continues): What were you doing there?

W: I had just logged in to see if your account was still active.

Question: M is suspicious that W just happened to be online at that very moment. He wants to have some mathematical measure of the following: Is it possible to calculate how often W must have been logged in every day or every week or whatever so that she happened to be online just at the same time?

(For example: The model may suggest she might be logged about half an hour everyday (or three times a week, or whatever) so that the incident occurred.

(I am aware that this may be a huge coincidence. I just want to know whether statistics has anything at all to say.)

Further question. (In my view, this should be workable somehow.) Exactly one year later the M logs in again to see whether he can now delete his account. Surprise, surprise! Another phone call from W! Leaving aside conspiracy theories (that W perhaps installed a spying software on M's computer, or whatever.)

Now M is really suspicious. Again: Is it possible to calculate how often W must have been logged in every day or every week or whatever so that she happened to be online just at the same time?

(Assume that at both times M was not aware that W was also online.)

• You would need additional information; in particular, how long M and W stayed on the site each time. W could have logged in and stayed in forever. – Peter Flom Jul 14 '18 at 12:12

As Peter Flom says in his comment, the statement is rather incomplete to perform an statistical analysis. However, since that just means that the problem is not properly modelled and you are asking about modelling, I'll give a try to finding a simple model.

In fact, since W's goal is to decide if M tells the truth, I'm going to assume that W is performing a test and that the null hypothesis of that test is that M tells the truth. I'm going to make more assumptions, but you can take them as examples and modify to assumptions that fit better your problem. We just need to model what happens if M tells the truth. That eases the problem a lot, because there is just one story he tells, but an infinite set of possible situations if he is not telling the truth.

Let's assume M's story is that he just entered the website once a year to try to delete his account - the times may not actually be completely random, but as long as they are independent from the times W logs in we can take them as uniformly random. Assuming that there is a window of time available to log in (he doesn't log in from work or while sleeping), and that the duration of each session is negligible compared to the time available in the day, the probability of M being logged in in a given moment is:

$$P(logged.in)=\frac{duration.of.session}{time.available}$$

For example, if M could log in at any moment in 8 hours each day and a session would take 6 minutes:

$$P(logged.in)=\frac{0.1hours}{365days/year·8hours/day}=3.42·10^{-5}$$

If W repeats the experiment, the probability of finding M logged in twice depends on whether both experiments are independent. If at least M or W have chosen at random the times in the two different years, then the probability of finding M online twice is the square of the previous probability ($1.17·10^{-9}).$

W could have stated an hypothesis test with:

$H_0$: M just logged in at one random moment each year for 6 minutes. $H_1$: M logged in for more than one random moment each year and/or did it for more than 6 minutes.

Taking as a contrast statistics the number of times M was caught when doing two tests, the p-value is the probability of meeting him twice assuming $H_0$, and it's tiny. Therefore, W should reject $H_0$ and accept $H_1$, that is, that M is not telling the truth.

Addition: from M's point of view

From M's point of view and provided that he has only logged in once a year, probabilities would be reversed. That is, assuming that W has logged once a year at a random moment (distributed independently of the moments M has logged in), the probability that those moments coincide with the moments M has logged in is tiny. There is a caveat that if M is actually trying to delete his account and not performing a test, p-value could be inflated - in other words, if we pay attention to a large enough number of phenomena, we will notice some unlikely coincidences even if they are meaningless. However, with such a tiny probability we can be rather confident that some assumptions of the model don't hold. Which one doesn't hold and what does it mean for M and W's relationship is a question we can't answer with the available data.

In fact, if this question is motivated by a real M and W I think you will get better advice from ips.stackexchange than from stats.stackexchange.

• Thanks. M can already sense that W's story is too much of a coincidence. What M wants is whether he can come up with a meaningful number as to how often should W be logging in to make this "coincidence" plausible. (Or it may the the case that given the available data, what is asked is not possible.) – blackened Jul 14 '18 at 16:34