We’re going to do some math today, and crunch some numbers to explain my experience with the iPhone 4 so far. We’ll be doing some statistics—I wouldn’t say this is really my area of expertise relative to some of my colleagues, but I should know enough well enough to string together a coherent argument.
We first identify the study population (say, all iPhone 4′s in Canada) and parameters for attributes we are interested in (we will need θ to be the proportion of these iPhone 4′s which are faulty/defective pieces of lemonware—in other words, the chance that you will receive a catastrophic failure of an iPhone).
We start by assuming that the parameter θ has a certain reasonable value (like 0.001, which says, on average, we think 1 in 1000 iPhone 4′s are defective in the study population). You should note that any numbers here are just guesses I’m making for illustration’s sake. Nowhere am I claiming these values or figures are real or confirmed or even remotely representative of the actual failure rates. Don’t sue me, Apple. Your legal expenses will be greater than everything you’d get bankrupting me.
Anyway, say this is the case. Then let the random variable Y be “we get a working iPhone 4 on the kth iPhone” where k = 1, 2, … (all positive integers). So then, the probability of the random variable Y taking on the realization of the plain integer y would be:
P (Y = k) = θ k – 1 (1 – θ)
You may recognize this as the geometric distribution and indeed it is. What we are saying is, there will be (k – 1) catastrophic failures before the kth iPhone is one that works. We also assume you’d stop driving all the way to the Apple Store to ask for and pick up replacements once you get a working one. I think this model is fair, as long as you stop replacing them eventually.
For example, if the first iPhone you get is fine, then k = 1 and we are done. The probability of this is P(Y = 1) = (1 – θ) (since θ to the power of 0 equals 1). Substituting the hypothesized value of θ in, you’d get that the chance of you getting a working iPhone 4 on the first try would be something like 999/1,000 or 99.9%.
Say your first one is a dud, but the replacement works out to be perfect. Then:
P(Y = 2) = (1/1,000)(999/1,000) << (the chance of a lemon) x (the chance of not a lemon)
= 999/1,000,000
= 0.000999
= 0.0999%
However, if you are me, then let k be greater than 2.
P(Y = k) = P(Y > 2) << strictly greater than 2
= 1 – [P(Y = 1) + P(Y = 2)] << this is the probability of everything (1) minus the prob’s of k = 1 and k = 2
= 1 – [999/1,000 + (1/1,000)(999/1,000)] << calculated above to be 0.999 + 0.000999
= 0.000001 << this is 1 in 1,000,000 if you don’t realize
That’s right folks. If the 1:1,000 failure rate is anywhere close to reality (and I’m not saying it is) then what I’ve gone through so far is one in a million. Of course, if the real rate is closer to 1% then it’d be like one in ten thousand, which is still really unlikely, though scaled back a bit.
Basically, the first iPhone 4 I received (after ordering on launch day and waiting three weeks for it to ship—nevermind the mishap where FedEx sent my iPhone to me, then to Vancouver, before sending it back to me) turned out to have a busted compass (GPS sensor was broken) and a busted gyroscope (a new sensor in the iPhone 4). I went to the Apple Store at Fairview yesterday to get it replaced—they didn’t have any stock so they had to order a replacement unit for me. I was told it would take three to five days to get there, but it arrived by 9 pm the same night. So, like a good boy, I drove down to the store again today to pick up the replacement. Just my luck: the replacement was another lemon! iTunes could not detect/connect to the replacement phone. Seriously. Two in a row, both lemons, when this phone is supposed to be of something really special/high-calibre.
Sidenote: you’d think someone at Apple would’ve thought of “let’s check these replacement units before they leave the door because it might make us look completely stupid if we replaced a lemon with another lemon”, but no. I can’t explain why (ask Apple, they might have a coherent reason, but I can’t think of any).
If I ever do get a working iPhone 4, it’ll be in the case where k > 2. If the next one works (i.e. the third one), then I think the probability of that (being a subset of k > 2) is something in the range of 0.000000999, or 9.99*10-9 if we stick with the 1 in 1,000 assumption.
Anyway, fired off an email to Steve Jobs and I wonder if he’ll ever respond. I asked him to put a few of his engineers on the task of figuring out the exact probability of getting my package incorrectly sorted by FedEx, having two sensors arrive as broken in the same iPhone, and then getting a lemon replacement unit that can’t be detected by iTunes. I added a bonus question for them to predict the probability that my next replacement unit would actually work as advertised.
It wasn’t all sarcastic though—I did mention the fact that I’ve often stood up for Apple’s philosophy of making high-quality stuff that’s worth the extra money, provided it works, but the bullshit attitude of the “Genius” girl at the Fairview Apple Store today and the impotent store manager (who could only offer apologies and “explanations” like “Well as you can see there’s nothing I can do about anything ever”) really made a sizeable dent in my opinion of Apple. Sure there’s nothing you can do, but how about taking my suggestion of “check the replacements before they are shipped out, before you call me, and before I drive down here” seriously, rather than rolling eyes?
Anyway, back to the math: either I really am lucky enough to win the inverse lottery 5+ times (FedEx shipping, attenuation not fixed, Sensor 1, Sensor 2, Lemon Replacement, and probably something else to come)—I was fortunate enough to have, in my one experience, observed something unbelievably unlikely—or the failure rate for these components is actually a lot higher, making these kinds of problems a lot more common than Apple would like. Surely, it must cost more to be playing these cat and mouse games with replacement units than to just employ better quality controls. Then again, it costs Apple next to nothing when I’m the one taking time out of my day and gas out of my car (or bus tokens out of my pocket) to pick up hardware replacements (or in my case, get head-faked again).
Hopefully I get a response from Steve Jobs and I can post the results here. What I really want to see is some sort of “I will look into it” from him. I still love Apple’s stuff, I just really hate my luck and MAYBE I’m starting to hate their quality (controls) and hardware replacement process.
PS: To the “Genius” girl working at the Apple Store rolling your eyes when I mentioned “You can’t keep my signed replacement receipt if this phone doesn’t work, by the way”, FFFFFFFFFFUUUUUUUU.