Hadrianus wrote:ThymineC wrote:Hadrianus wrote:We? Who is "we" ? Or are you using the "Royal we" to describes yourself?
He's talking about you. You said:
Hadrianus wrote:On earth you see that the poorer and less educated a country becomes the more the population is inclined to believe all sort of nonsense.
Populations tend to be made up of a fair number of people.
Please elaborate! I don't think I'm getting what you are trying to say.
Assume the existence of two individuals, X and Y, who are from different countries, XLand and YLand. Let's say that individuals from YLand have a higher
inclination to be superstitious - say people from XLand have "40% probability of being superstitious" (whatever that means) and people from YLand have 60% probability.
Now, odds are that Y will be superstitious, and X won't be. But that could very easily not be the case either - maybe neither are, both are, or X is and Y isn't. There's a 24% chance of the first case, 24% chance of the second case, and 16% chance of the third case.
But now let's consider this same argument being applied over a population. Assume XLand and YLand both have populations of 1000 people. We will also assume that people form their beliefs independently of what others in their country believe. In that case, we can model the number of superstitious people in XLand ~ Bin(1000, 0.4) and YLand ~ Bin(1000, 0.6).
What is the probability that YLand has more superstitious people than XLand? In fact, let's change this to "what is the probability that XLand has at least as many superstitious people as YLand?" We can posit two random variables based on these binomial distributions:
A <- Bin(1000, 0.4)
B <- Bin(1000, 0.6)
And then what we essentially want is the probability that A >= B i.e. P(A - B >= 0)
According to
the top answer of this post, we find that the Hoeffding bound of P can be expressed as:
P(A - B >= 0) <= exp( -0.5*1000*(0.6 - 0.4)^2 )
We know that if P represents the probability that XLand has at least as many superstitious people as YLand, then Q = 1 - P must give the probability that YLand has more superstitious people than XLand.
Therefore:
Q = 1 - exp( -0.5*1000*(0.6 - 0.4)^2 )
= 0.99999999793
That means there's a 99.999999793% chance that YLand will contain more superstitious people than XLand given the stated probabilities of superstitiousness among individuals and the given population of each country.
Furthermore, we see that as the size of the populations get larger, the Hoeffding bound of P shrinks to zero and Q tends to 1 (certainty). I.e.:
lim[n->infinity](1 - exp( -0.5n(0.6 - 0.4)^2 )) = 1
=> lim[n->infinity](Q) = 1
Basically, stating that the "inclination of individuals within country X is to be more superstitious" is, for all practical purposes, the same as stating "there are more superstitious people in country X".
QED.
Gonna get dinner.
Edit: Accidentally wrote one word in place of another in a way that might have caused offence. Corrected.