We might not talk again

A question crossed my mind recently concerning my inclination to reduce my attention to notifications from any social media on my phone.

For years, I would be intermittently checking my phone rather subconsciously, though the reasonable part of me knew well that there is no point checking my phone at too high a frequency. Nevertheless, I never seriously figured out, or attempted to ballpark this frequency. This question I came up with, or perhaps an oversimplification of my long-lasting concern, goes as follows: Suppose I send a message to each of my $N$ friends. What is the expected time needed to get the first response back? Likely the days of doing mathematical exercises are distant from me and it was not until the past two weeks that I came to realize the simplest variant of this problem is an exercise of order statistics.

Putting it more formally, suppose the probably that friend $i$ responded between $t$ and $t + dt$ , counting from the moment I had sent out the initial message, is $f(t)dt$ . $P(T_i < t) = \int_0^tdt', f(t'):= F(t).$ The probability that the arrival time of the first response is smaller than $t$ is naturally $1 - P(\min T_i > t) = 1- P(T_i > t) =1- [1-F(t)]^N$ and we have the probability density $Nf(t)[1-F(t)]^{N-1}$ and $\mathbb E[T_{\min}] = \int^{\infty}_0 dt\, [1 -F]^N$ .

The precise value would naturally depend on the choice of $f$ , a felicitous choice being the family of Gamma distribution. The framing has been pretty standard up to this point, and the conclusions were scarcely exciting. What I really found formidably intriguing was a relatively less trivial extension: suppose I lost connection with one of them, say friend $i=1$ , and in light of how the fallout took place there was a probability $p$ that I also lost friend $j$ . With the introduction of $p$ , it was hardly troublesome to deduce that $P(T_{\min}>t) = [1-pF]^{N-1}$ and observe $P(T_{\min}=\infty)= (1-p)^{N-1}>0$ .

Potentially, no message might come back.

I was never certain how other people conceived of it when they casually uttered statements such as “We might not talk again.” Perhaps, it was literary to most of them, and yet, to me, it could be most literal. All in all, it was not forbidden, and, as nature would often demand, hence its inevitability would be hardly unconventional.

We might not talk again

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