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# Oldest Probabilistic Evidence of Divine Providence

You don't have to worry that we're changing the focus of our server. Let's look at one of the oldest mathematical or probabilistic proofs, dating back to 1710. Dr. John Arbuthnott systematically studied the numbers of births and their genders for the past 82 years. He came to the conclusion that larger numbers of male births than female births are not the result of chance but of divine providence. How did he reason?

According to Dr. John Arbuthnott, one of the most remarkable proofs of divine providence (→ An Argument for Divine Providence) belongs to Mother Nature, maintaining balance between the number of men and women. This ensures the survival of the human species. Every man finds a woman of suitable age. This balance is not the effect of chance but evidence of divine providence pursuing a good goal – the preservation of the human generation.

Dr. Arbuthnott studied records of London registries from 1629 to 1710, a period of 82 years, and found that in each year, more boys were baptized than girls. You might be wondering what's so unusual about this.

## Could you Flip the Same Side of a Coin 82 Times in a Row?

Let's consider that the probability of a boy or girl being born (baptized) is absolutely the same (in reality, it is not, as we now know with certainty), i.e., 50:50 or, mathematically expressed, `0.5` (`50%`). It is the same as flipping a coin that has two sides with an equal probability of landing on either one.

Let's say a boy = heads and a girl = tails. Now we take 82 coins and toss them into the air. What is the probability that after landing on the ground, all coins will show heads (boys / will be born or baptized as boys)?

The total number of possibilities that can occur when tossing 82 coins is:

`282 = 4,835,703,278,458,520,000,000,000`,

which is almost 5 sextillion → names of large to astronomical numbers. There is only one single possibility(!), that all coins will show heads. The probability of such an event is one in about five sextillion (easily surpassing numerical lotteries such as Keno, where there is also a huge number of possible combinations), expressed as a decimal number:

`1 ÷ 4,835,703,278,458,520,000,000,000 = 0.000000000000000000000000206795`.

The same result is obtained by tossing a single coin 82 times in a row. The probability that heads will appear 82 times in a row – translated into our example: the probability that in the next 82 years, more boys will be baptized than girls every time (assuming they can be born with the same probability of 0.5) – is:

`0.582 = 0.000000000000000000000000206795`.

This is just a shortened notation of the so-called binomial distribution, which represents the probability that an event will occur exactly k times in n attempts with the probability of the event p.

## Conclusion

From a probabilistic perspective, this is an almost impossible event (= zero probability), so Dr. Arbuthnott concluded that more boys are simply born. In the language of mathematical statistics: he rejected the hypothesis that boys and girls are born with the same probability.

The proof also had a moral or ethical dimension. According to Dr. Arbuthnott, polygamy goes against natural laws, justice, and the reproduction of the human generation; if the numbers of men and women are roughly equal, and one man takes twenty women, then nineteen men must live in celibacy, which is supposedly against nature. The proof's conclusion sounds somewhat amusing: it is unlikely that twenty women would be fertilized by one man as well as by twenty different men.

Source: École normale supérieure (ENS). John Arbuthnott: An Argument for Divine Providence [online]. ENS [cited 2014-03-22]. Scanned original document, PDF format, English language. Available from: <http://canoe.ens.fr/~ebrian/s1h-dhsrb/1710-arbuthnot.pdf>.

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Based on the original Czech article: Pravděpodobnostní důkaz o božské prozřetelnosti.