Verbum Sat Sapienti Est

So we’ve just bought a couple new iPhone 4S handsets and re-upped with AT&T for two years.  We got a little bit of upsell pressure from the (otherwise pleasant and capable) AT&T salesperson to take out their phone insurance, but I demurred, thinking I’d opt for AppleCare+ since it now includes accidental damage protection.  I’d read that there was some sort of December 15 deadline to buy Applecare+ over the phone, so I figured we’d just call and add it.

I was wrong; that December deadline was only for folks who’d bought their phones before the advent of the new service, which we hadn’t.  Instead, we’d need to make a Genius Bar appointment and drive to the nearest apple Store (the Mall of Georgia) to have the phones inspected by a Genius if we want to purchase coverage.

So, the prospect of driving to the mall on the last weekend before Christmas gives me some pause.  I also read this piece over at The Mac Observer, which is more to think about.  The AT&T plan clearly makes no sense; although it covers theft, which Applecare doesn’t, it’s $7 a month, per handset, and replacements are $199.

Since “Smart Choices” is sitting on my desk (full disclosure: it’s 100% unread), and this is the sort of problem that could be approached  with a structured decision, let’s give it a shot.  We have a little data: I’ve had an iphone of some stripe for 4 years and have only had one headset replaced under warranty (which was arguably an indulgence from Apple for a wear item and not a true warranty claim).  My wife’s had an iPhone for 2 years without a claim (there was that incident with a cat bowl, but I still have some silica gel left over from a successful home-remedy.)  So, resisting the gambler’s fallacy, we can assume that the probability of getting through our 2-year contract without incident (since there’s a hardware warranty for 12 months, anyway) is fairly high.

Assumptions:
Applecare+ is $99, per phone, with $49 replacements for damaged handsets (up to two, per policy)
The AppleStore is an hour away, and it’s worth $30 per hour for me not to drive there.
Apple maintains their policy of offering $200 refurbs for damaged out-of-warranty phones.
If we break 4 handsets, the distribution will be 2, per policy

This gives us the following cost breakdown:

So, the optimal choice changes somewhere between one and two replacements?  Not once we add what we know about our particular phone-replacement history.  Ideally, I’d model the probability of breakage as some sort of Poisson process with a long tail, in which it would be possible, though unlikely, for us to break a dozen phones or more. For simplicity, though, let’s just say there’s a 2 in 3 chance we don’t break a phone, and there’s an evenly-decaying probability of 1-5 mishaps.  If we multiply these odds into the cost estimates, above, we get:

So, given this estimated risk profile, winging it without insurance is the better plan (by a factor of two).  How sensitive is the model to our assumptions about risk?  What if we impose at least one broken phone on the risk profile?

Even if we assume at least one mishap, the choice between the plans is essentially a wash and I have my Saturday free.

Is there a better way to do this analysis?  Please let me know what you think in the comments.