Every pixel on the panel must work, otherwise the whole panel is useless. Now imagine you have one broken pixel on average for every million pixels. Apple Watch screens have 100k pixels so on average 9 out of 10 screens test OK and can be used to make a watch. But the iPhone has about a million pixels, so you end up throwing away about half of all the screens you make.

More pixels = lower yield.

Probabilities works out even worse. For million pixels roughly 37% of screens would turn out OK.

Somethings off. It should work like this:

  p = 1e-6
  n = 10^(1:7)

  frac_accepted = (1-p)^n

Results:

  num_pixels frac_accepted
      1e+01     0.9999900000
      1e+02     0.9999000049
      1e+03     0.9990004993
      1e+04     0.9900498288
      1e+05     0.9048373728
      1e+06     0.3678792572
      1e+07     0.0000453997

This assumes each failure is independent. Often a single failure will cause many bad pixels on the same screen. So you might have 98 good screens 1 with one bad pixel and one with 5,000 bad pixels when the per pixel odds say you are going to end up with 1 good screen if you are lucky and 99 bad ones.

I simulated 100 screens with different numbers of pixels where the probability the first pixel is bad in a screen is p0 = 1e-6 and the presence of each bad pixel increases the probability of the next one by the same amount (ie probability for a second bad pixel is 2e-6 and for last pixel is p0*number_already_bad_px). A screen was accepted if it had 0 bad pixels. The results look pretty much the same:

  num_pixels frac_accepted
       1e+01          1.00
       1e+02          1.00
       1e+03          1.00
       1e+04          0.98
       1e+05          0.89
       1e+06          0.32
       1e+07          0.00

Of course this was just a convenient scheme to simulate. The greatest number of bad pixels was 86k for the 10 million px screen (so chance the last pixel was bad was ~9%). It does show that the independent pixel model isn't necessarily very far off though.

You messed up the math. You need to count (# number of pixels that where bad) / (total number of pixels produced) not simply call the odds for the first bad pixels as the rate of bad pixels.

You are saying to adjust p0 so I get that the frequency of bad pixels = 1e-6? That makes sense but it would probably now depend on what size screens are being considered... not sure if I'll go back to it.

I didn't actually calculate the probabilities, it was just my off-the-cuff estimates. 37% could be 'about half' :P

>Every pixel on the panel must work, otherwise the whole panel is useless

And that is not so far fetched. The most high yield semi processes can give you defects in single digits for a fully exposed 300mm wafer.

The challenge here is that they will have to be grown on a substrate different than silicon for anything bigger than apple watch screen.

Growing inorganic semiconductors on glass, organics, non-crystalline materials is what is hard

Could smaller adjacent panels be coordinated to appear imperceptibly as if they’re a single panel?

Potentially, that's how led billboards are assembled.

I guess it would be about the ratio of the price for the screen compared to the device. Smartwatch at 600$ with a small screen vs. a smartphone at 1000$ but with a 10x bigger screen. With 10 smartwatches I would have 6000$ to spread the additional cost over.

[edit] typo

Assuming the same (as current) prices for iPhone and smartwatch, that would be the same ratio though if we divided by the current models, so it doesn't tell us much regarding how might the new kind of screens affect price.

Only something that took into account the relative cost of each type of screen would (and, if screen area is a non 1x linear factor of cost, that too).