## Monday, April 04, 2016

### The Amazon parcel problem

Alice and Bob are both signed up to Amazon Prime and each is expecting a parcel to be delivered to their respective apartment the very next day. The Amazon delivery person will arrive sometime in the ten hours from 9 am to 7 pm, but there's no more definite information. Let's say delivery has a uniform probability distribution.

Both Alice and Bob plan to be at home to collect the parcel, but each also wants to spend an hour in the bath from where, unfortunately, you can't answer the door.

Alice says,
"The chances of the Amazon guy turning up in any of the 10 hour-long slots is exactly the same. It therefore doesn't matter when I take my bath, I always have just a 1 in 10 chance of missing the parcel."
Bob says,
"I'll be taking my bath at 9 am. If I do that I've got a 9 in 10 chance of the Amazon guy coming later than that.

"But suppose I'm stupid enough to delay the bath till, say, 5 pm. Sure if the guy already came, I'm good. But assuming the parcel hasn't yet arrived, there's now a 50% chance I'll be in the bath when it arrives. I'd be insane to bathe then.

"It's always better to take a bath early when an Amazon parcel is due."
Who's right?

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Update: (Tuesday, 5th April, 2.15 pm).

I'm  more interested in why Bob's argument might ever be considered psychologically plausible.

Let's say we get to 2 pm and the Amazon guy hasn't come. Then the probability per hour of the parcel arriving in the next five hours has doubled from 10% to 20%.

Definitely riskier to take that bath!

This assumes that Bob forgets the (on average) half the days that the parcel did in fact come earlier - he collected it or it arrived between 9am and 10am and he missed it, being in his bath.

But people have a past-future asymmetry; they forget the past but worry about the future. Early deliveries don't give much cause for worry; late deliveries generate increasing anxiety as the day progresses.

People remember things like that - the late days come to predominate in memory. And they have the higher per-hour probability distribution for when the parcel will eventually be delivered.

'Best to bathe earlier'.