Ran a simulation with N = 20000.
Here's the graph, as requested.
And here's the code.
EDIT: As a sidenote, the average percentage of infected was 71.5% after twenty simulations. Someone else can strive for statistical rigor, it's daddy's B5 time now.
I'd like a small clarification. When selecting a victim, is it possible to select yourself? The OP isn't entirely clear on that. Specifically "If they target another person who knows the information (even themselves)" seems to contradict "by randomly choosing someone else
to inform." Until I hear anything else I (and the above code) will just assume it's possible to target yourself.
I'd like to independently confirm snail's result, I get the same result with a similar probability model. Snails did have a minor inaccuracy there but the effect becomes insignificant as sample size is increased (The chance of targeting yourself needs to be addressed, however the above is corrected.)
The percentage of total infections seems to approach 75.98% as the sample size is increased. (Even N = 200 gets over 75.5%)
I expect there's a bug in my above program though. The rate of infections should be close to 2^n with small enough n, so why am I seeing a rate of n? Needs checking.
EDIT: EDIT: EDIT:
Found the bug. Corrected the above code (The line or i=1:count_a(end); lacked the (end) which caused it to do just one infection attempt per round). Replaced the image too. Now the result seems much more in line with the probabilistic estimate. Also, the simulation ran significantly faster now that it is bug free, so this time I ran it with a population of 2000000. And the percentage of infectees is 76.03% in simulation.
Maturity is knowing you were an idiot in the past.
Wisdom is knowing that you will be an idiot in the future.
Common sense is trying not to be an idiot now.