'Waiting in' for the repair man/courier, etc. - when IS the best time to nip out?

I realise this is basic probability theory, but almost no one gets this right, so I thought I'd put it in plain English...

The situation. You're expecting a package delivered (and needing signing for), or perhaps it's a repair man. Either way, you know it's happening today, you just don't know when. You may even have taken the day off work to accommodate this.

So you're waiting in. It might be 10 minutes. Or it might be 10 hours (a 8am-6pm window). You just don't know.

Yet you want to either:
  • nip to the local shops for milk
  • have a shower/toilet/whatever
  • some other task which only takes a few minutes but which will mean you're unavailable
When should you do this? Is there a best time according to the laws of probability?

Common sense says that it's random when the courier or repair man comes, so it doesn't make any difference when you take your break.

But in fact, although the arrival is indeed random, the fact that your own unavailability is under your control means that actually, yes, there IS an optimum time to break - and it's as soon as possible.

You'll realise why after a moment's thought. If you're going to be unavailable for (say) 15 minutes and the courier or repair man is coming in the next 10 hours, that's only a 1 in 60 chance of missing them, i.e. 1.6%.

But if you delay and wait until (say) 1pm and they still haven't arrived, then your 15 minute break has a 1 in 20 chance of overlapping, i.e. 5%. Leave taking your break until 5pm and it's one in four, or 25%!

In reality, it depends why you need to be away and how urgent it is, since balanced against the probabilities above is the scenario where you wait in until the chap has come and then do whatever it is you need to do.

But the thought exercise was one that always intrigues me, since the situation happens surprisingly often in my life. I work from home and there's always someone coming or something being delivered, yet at some point I need to disappear into the toilet/shower, at some point I need to grab some exercise, at some point I need to head into the garage to sort out the laundry.

And it turns out that the best time to do any of these is as soon as possible and not to worry about waiting....!

Is it just me that ponders these statistical mysteries of daily living??

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