OK, let’s clarify this.
10 objects with 50 bits of FI each are not, in any way, 500 bits of FI.
10 objects with 50 bits of FI each are 500 bits of FI only if those 10 exact objects are needed to give some new defined function.
Let’s see the difference.
Let’s say that there is a number of possible functions in a genome that have, each of them, 50 bits of FI.
Let’s call the acquisition of the necessary information to get one of those functions “a success”.
These functions are the small safes in my example.
The probability of getting a success in one attempt is, of course, 1:2^50.
How many attempts are necessary to get at least one success? This can be computed using the binomial distribution.
The result is that with 2^49 attempts we have a more than decent probability (0.3934693) of getting at least one success.
How many attempts are necessary to have a decent probability of gettin at least 10 successes, each of them with that probability of success, each of them with 50 bits of FI?
Again, we use the binomial distribution.
The result is that with 2^53 attempts (about 16 times, 4 bits, the number of attempts used before) we get more or less the same probability: 0.2833757
That means that the probability of getting 10 successes is about 4 bits lower than the probability of getting one success. The FI of the combined events is therefore about 54 bits.
Why is that? Why do probabilities not multiply, as you expect?
It’s because the 10 events, while having 50 bits of FI each, are not generating a more complex function. They are individual successes, and there is no relationship between them.
That’s why the statement:
10 objects with 50 bits of FI each are not, in any way, 500 bits of FI.
is perfectly correct. Those ten objects have 500 bits of FI only if, together, they, and only they, can generate a new function.
In terms of the safes, solving the 100 keys to the samell functions generates 100 objects, each with 1 bit of FI. But finding those 100 objects does not generate in any way 100 bits of FI, because the 100 functional values found by the thief have no relationship at all with the 100 bit sequence that is the solution for the big safe.
I hope that is clear. We can rather easily find a number of functions with lower FI, but their FI cannot be summed, unless those functions are the only components that can generate a new function, a function that needs all of them exactly as they are.
Please, give me feedback on this point, before I start examining the example of affinity maturation in the immune system.
This is not only to Swamidass, but to all those who have commented on this point.
By the way, I was forgetting: using the binomial distribution, we can easily compute that the number of attempts needed to get at least one success when the probability of success if 1:2^500 (500 bits of FI) is 2^499, with a global probabilty of 0.3934693.