Randomness and populations

king_of_inismac

I've always been particular interested in the maths behind randomness.

So I was considering the random way in which gnder is assigned during conception.

Since each assignmemt is independent and random, shouldnt we expect for there to be significant variances in the male/female divide amongst populations worldwide?

Now I know in a *fair* system, each option is equally likely to happen so male or female are equally likely, but isnt it the *gamblers fallacy* to say we should expect to get a 50/50 result from even a fair system.

Now, i know the polulation divide isnt exactly 50/50 male female but its pretty damn close. Why arent we suprised that its not 60/40 or even 53/47?

Just a musing!

hivizman

king_of_inismac wrote: »

I've always been particular interested in the maths behind randomness.

So I was considering the random way in which gnder is assigned during conception.

Since each assignmemt is independent and random, shouldnt we expect for there to be significant variances in the male/female divide amongst populations worldwide?

Now I know in a *fair* system, each option is equally likely to happen so male or female are equally likely, but isnt it the *gamblers fallacy* to say we should expect to get a 50/50 result from even a fair system.

Now, i know the polulation divide isnt exactly 50/50 male female but its pretty damn close. Why arent we suprised that its not 60/40 or even 53/47?

Just a musing!

If the size of the population is n and the probability of a person being male is p, then the distribution of the number of males in the population will be a binomial distribution with mean np and variance np(1-p). For example, if the population is 1 million, and the probability of a person being male is 0.5, then the number of males in the population will follow a binomial distribution with mean 500,000 and variance 250,000. As the variance is the square of the standard deviation, this implies that the standard deviation of the distribution is 500. This is the key to its being unreasonable to expect the proportion of males to deviate much from the mean for large populations.

For large values of n (here, "large" can be as few as about 30 if p is close to 0.5), the binomial distribution is approximated by the normal distribution with the same mean and standard deviation. We would expect about 95% of the possible values in a normal distribution to lie within two standard deviations of the mean, so in a population of 1 million, the probability of having 1,000 (2 x 500) males more or less than the expected value of 500,000 is around 5%. For more than 53% or less than 47% of the population to be male, we are talking of 60 standard deviations away from the mean, and the probability of that is approximately zero.

In practice, the probability of someone in a population being male is not necessarily exactly 0.5, because of differential perinatal and infant survival rates, tendency of women to live longer, and special circumstances such as wars (which tend to lead to more men than women dying), and economic development (more expat males working in UAE, for example, than females). Also, it appears that on average 105 males are born for every 100 females.

This article has some interesting data on gender balances in different countries, and it shows that there are indeed significant differences from one country to another.

king_of_inismac

Interesting reply, but why should we EXPECT a binomial or normal distribution? Isnt a 53/47 split equally as likely as a 50/50 split in a truely random system?

Grudaire

The argument is that each conception is an independent event that can take either of two values (Male/Female)

The Central Limit Theorem says that a sum of n independent Random Variables with any distribution has a normal distribution for large enough n.

The key thing is that there is a lot of independent events,

hivizman

Cliste wrote: »

The argument is that each conception is an independent event that can take either of two values (Male/Female)

The Central Limit Theorem says that a sum of n independent Random Variables with any distribution has a normal distribution for large enough n.

The key thing is that there is a lot of independent events,

The assumption is that the biological gender of one person in a population is independent of the biological gender of each other person in that population. This assumption of independence is in strict terms untrue because of identical twins and other factors, although there is no reason to believe that there are more likely to be twins of one gender rather than the other.

Suppose that it's equally likely that a person will be male (M) or that the person will be female (F). Then with one person there are two possible outcomes: M (with probability 0.5) and F (with probability 0.5).

With two people, the first person can be male or female and also the second person can be male or female. This gives four equally possible outcomes, MM, MF, FM and FF, each with probability 0.25 - this is so because the gender of the first person is independent of the gender of the second person, so the probability of each combination is 0.5 x 0.5 = 0.25). But in terms of the population, there are only three cases:

All male (MM), probability 0.25
One of each (MF or FM), total probability 0.5
All female (FF), probability 0.25)..

For three people, again the first person can be male or female, with equal probability, the second person can be male or female, with equal probabiility, and the third person can be male or female, with equal probability, and the gender of any person is independent of the gender of any other person. This gives eight combinations: MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF, each with a probability of 0.5 x 0.5 x 0.5 = 0.125. There are four combinations:

All male (MMM), probability 0.125
Two males, one female (MMF, MFM, FMM), probability 3 x 0.125 = 0.375
One male, two females (MFF, FMF, FFM), probability 3 x 0.125 = 0.375
All female (FFF), probability 0.125

You should be seeing the pattern by now - and this pattern is simply the binomial distribution with p=0.5.