Steve
Crawford’s idea of reproducing the calculations shown in the film is an
intriguing one. Unfortunately, the model chosen by Steve is formally not
correct for the situation. He seems to assume that Poisson’s
distribution has a finite support (i.e. in our case, the outcomes are
limited to the 12 persons at the outpost) whereas the theoretical set of
outcomes is infinite (from 0 to infinity.) The numbers he got apply to a
population infinitely large, i.e. it could be applied to the whole of
Earth (if we assume “infinity” as a very large number so that the
probability estimates are “close” to the theoretical ones.)
But what
happens if we want to use Poisson’s model to the limited population of
the outpost? How far off are Steve’s estimates with this approximation?
The goodness of the approximation depends on the infection rate.
The correction
can be calculated by using the Poisson formula again, for the
probability of events larger than 12. For the two extremes of the
infection rates, 1.116 and 1.371, these probabilities are 2.38*10^(-10)
and 2.73*10^(-9), i.e. of the order of 1 in 1 billion. This means in
practice, that the model’s calculations are not affected in practice by
the error.
What’s
interesting is to calculate the “effective” size of the population to
which the model can be applied and still obtain reasonable
probabilities. By assuming the largest infection rate of 1.371, the
probabilities are as follows:
Minimum
number of
infected
persons |
Probability |
1 |
74.61% |
2 |
39.81% |
3 |
15.95% |
4 |
5.05% |
5 |
1.31% |
6 |
0.29% |
7 |
0.055% |
So we see that
the probability decreases quite steeply. So, even if Blair didn’t know
much about the rules of probability to apply the “trick” reported by
Steve, it would have been enough to apply Poisson’s formula up to 5 or 6
to already get a good estimate.