6

In "I, Borg," while working on the invasive program designed to destroy the Borg, Geordi tells Captain Picard that within a few months the entire Borg Collective should be wiped out. How could he possibly know this? The Federation couldn't have any knowledge of the size of the Borg at this point. And even if they did, how could they possibly know how quickly the program would take to work? Geordi is smart, but I don't recall him owning a crystal ball.

What is the canon answer to this?

7

Exponential growth is really quite powerful: even a simply doubling from one once per day would get you to over a billion in one month, and 2,475,880,078,570,760,000,000,000,000 (2 billion billion billion!) in three months.

Even if infected Borg are eventually detected because they become "broken", and then disconnected from the collective, there would still be a massive growth - as long as each infected Borg can subsequently infect more than one other.

4

If you know the maximum speed of subspace communications and the throughput (bit-rate) at that speed, the size of the galaxy and the infection rate, you can extrapolate a rough date by which all the Borg should be infected. This is much like the rough calculation Spock (and later Dax) made about the number of tribbles on space station K7.

SPOCK: They seem to be gorged.

BARIS: Gorged? On my grain?! Kirk, I am going to hold you responsible. There must be thousands of them.

KIRK: Hundreds of thousands.

SPOCK: One million, seven hundred seventy-one thousand, five hundred sixty-one. That's assuming one tribble multiplying with an average litter of ten, producing a new generation every twelve hours over a period of three days.

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    "One million, seven hundred seventy-one thousand, five hundred sixty-one" < is a very specific rough calculation... – HorusKol Jun 28 '17 at 2:53
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    @Tim You're forgetting the tribble that you started with. Each generation is a power of 11. 1,777,561 is 11^6. – Kyle Jones Jun 28 '17 at 5:16
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    @HorusKol: That number could be a (precise) outcome of a calculation which is based on estimated numbers; therefore still being an estimate. – Flater Jun 28 '17 at 10:39
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    @HorusKol: It's not illogical. Keep in mind, the calculation of tribble spawning is of an exponential nature, which can very quickly devolve into non-trivial numbers. Example, I put $100 in my bank account, at 10% compound interest. How much do I have in my account after 10 years? $259.37. I used multiples of 10, and the answer is still unexpectedly precise and complex. While my initial values all had a low precision, the outcome of the calculation has a high(er) precision. Yet; the outcome of my calculation can only be as accurate as the input. It can never increase. – Flater Jun 28 '17 at 14:58
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    @HorusKol: Spock even specifically mentions how he estimated it; what the calculation is based on. Spock has no issues in doing mental arithmetic (hence the incredibly precise answer), but he cannot accurately tell you how many tribble there were to begin with. No one knows, it is not a matter of being proficient at math. The only reason why you would say "around 1.7 million" is if you were either incapable or indifferent to do the whole calculation. Spock is not estimating the calculation, he is estimating the numbers that are put into the calculation. And that is a big difference. – Flater Jun 28 '17 at 15:09

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