Permutations, Probability and the Existence of God

Photo courtesy Wikipedia

One evidence for the existence of God is the existence of life itself.

The popular scientific theory (which is taught in schools) that life happened by accident is widely professed by scientists, professors, etc. But strictly by the numbers, the probability of the existence of life in the entire universe is extremely low. In fact it would be crazy to think of it as anything but a miracle. I will attempt to show a short proof in the following article.

To understand the argument we need to cover some definitions:

Probability

Example:

Let's say we have a dice with six sides numbered one through six.
What is the probability that you will roll a "1"?
The answer is 1/6 or 16.67%.
 
The formula:

Probability　＝	Number of things that actually occur
	Number of things that are possible

Permutation

Example:

Let's say you have two six-sided die. What are all the possible permutations for rolling the dice where order matters (e.g. 2,1 is considered distinct from 1,2)?

1,1	1,2	1,3	1,4	1,5	1,6
2,1	2,2	2,3	2,4	2,5	2,6
3,1	3,2	3,3	3,4	3,5	3,6
4,1	4,2	4,3	4,4	1,5	4,6
5,1	5,2	5,3	5,4	5,5	5,6
6,1	6,2	6,3	6,4	6,5	6,6

There are 36 or 6² possible unique orderings or permutations.

What if we add a third die?

1,1,1
1,1,2
1,1,3
. . .
6,6,4
6,6,5
6,6,6

There are 216 or 6³ possible orderings or permutations.

Here's the formula:

Permutations in an ordered set = n^m
Where n is the number of possible values per element.
Where m is the number of elements in the set.

Putting Permutations and Probabilities Together

We can answer some interesting questions using permutations and probabilities.

Example:

What's the probability of picking 6 random letters from the alphabet; say from 6 bags with letter tokens; and randomly spelling the word CHEESE?

First, we calculate the number of possible outcomes:

26⁶ = 308,915,776

Next write down the number of correct outcomes; There's only one way to spell CHEESE that I know of.

Finally, calculate the probability:

1/308,915,776

Supposing you could draw and test the 6 letters once per second. How long could it take you to randomly draw and spell the word cheese correctly under these terms?

It could take: 308,915,776 / (60x60x24x365.25) = 9 years, 288 days

DNA is necessary for life

DNA is like a blueprint for a plant or an animal. If DNA base pairs are not in the correct sequence, it cannot be used to create life. In this light, it is safe to say that one of a number of correct sequences is necessary for life as we know it.

Below is a partial breakout of the human genome that I got from wikipedia.
http://en.wikipedia.org/wiki/Human_genome

Chromo-some	Length (mm)	Base Pairs	Valid Variations
1	85	249,250,621	4,401,091
2	83	243,199,373	4,607,702
3	67	198,022,430	3,894,345
4	65	191,154,276	3,673,892
5	62	180,915,260	3,436,667
6	58	171,115,067	3,360,890
7	54	159,138,663	3,045,992
8	50	146,364,022	2,890,692
9	48	141,213,431	2,581,827
10	46	135,534,747	2,609,802
11	46	135,006,516	2,607,254
12	45	133,851,895	2,482,194
13	39	115,169,878	1,814,242
14	36	107,349,540	1,712,799
15	35	102,531,392	1,577,346
16	31	90,354,753	1,747,136
17	28	81,195,210	1,491,841
18	27	78,077,248	1,448,602
19	20	59,128,983	1,171,356
20	21	63,025,520	1,206,753
21	16	48,129,895	787,784
22	17	51,304,566	745,778
X	53	155,270,560	2,174,952
Y	20	59,373,566	286,812
mtDNA	0.0054	16,569	929

Each DNA strand is made up of 4 base-pairs, which can be put in any order, although there are specific rules such as each chromosome is made up of some number of genes, each gene is made up of a sequence of proteins which are made of sequences of base pairs. The gene is typically ordered with a protein called a "start codon" and ends with a "stop codon". There are a lot of other details which won't be covered here. but needless to say, DNA sequencing is pretty complex and specific.

So if I was to test the probability of finding a correctly ordered sequence for say, chromosome #22 "at random" I would proceed as follows:

First, calculate the number of possible combinations of base pairs:

4^51,304,566 = 3.6395x10^30,888,426

Now take the number of valid sequences and divide by the possible values to get the probability:

745,788 / 3.6395x10^30,888,426 = 2.0491x10^-30,888,421

This number is so small as to be unfathomable by the human mind. Happening upon even one sequence of this size in the correct order, is in and of itself a miracle.

A Simple Proof that Life is not an Accident

But we're talking about life, not evolution right? Nobody's saying that man "suddenly appeared". You're jumping the gun. Right. So after the "big bang", these things are supposed to have happened "very slowly" over a long, long period of time. Plus there are zillions of stars and planets in the universe, life was bound to happen on one of them eventually... Right? Right? Let's consider this point of view.

According to wikipedia, the smallest known cellular organism (as far as DNA goes) is Candidatus Carsonella ruddii, with only 160,000 base pairs. But scientists also lean toward the possibility of much smaller organisms as the first form of life. The smallest known quasi-living things are called circoviruses, which apparently can have as little as 2,000 base pairs. Whether they are alive is debatable, but for this argument, to make it interesting, we'll base our proof on a tiny DNA sequence like that of the circovirus.

Many theories base the origin of life on having the right ingredients in the right place at the right time and then adding energy in the form of electricity such as lightning, or direct sunlight, or thermal energy. Many experiments have been done with ambiguous results. For our approach we don't care where the ingredients or the energy comes from just that the perfect and ideal conditions already exist.

Modern estimates put the number of planets in the universe to about 10²⁴, but for argument's sake, let's say you have 1 centillion (10³⁰³) planets in the universe. Additionally, let's say that you have 1 centillion "hot-spots" on each planet already with DNA particles in place, energy, etc., simultaneously trying 1 centillion DNA permutations per second. Furthermore, the only goal you have is to produce one circovirus DNA set with 2,000 correctly-ordered base pairs.

The math looks like this:

4²⁰⁰⁰ (possible base pair permutations) / [10³⁰³ (planets) ｘ 10³⁰³ (hot spots) ｘ 365.25x24x60x60 (seconds/year) ｘ 10³⁰³ (permutations per second) ] = 4.144x10²⁸⁷ years

Given that the current estimated age of the universe is 13 billion (1.3x10¹⁰) years, even with the insanely contrived conditions I just proposed, the universe isn't remotely old enough to have "spontaneously" produced life in this manner. So, unless you believe in miracles it is very, very, very (add about 10²⁸⁷ "very"s here) unlikely that DNA-based life could have spontaneously occurred in our universe.

There is also one very important point I haven't made yet. Assuming that somehow all of the right ingredients for DNA landed in the right place at the right time (however improbable), AND in the perfect environment (nutrients, heat, moisture, etc.) AND all of the DNA sequences just happen to be in the correct order (very unlikely, refer to the treatment above); Even if you had ALL that, can you really say that life would occur? Would a million lightning bolts be enough to do it?

In other words, without God, the chance that life has occurred "by chance" is ludicrous.

確率と順列と神の存在

－　草案です　－

神の存在の証拠の一つは生命の存在です。

 人気の（学校で教わっている）科学者の理論では地球での生命の存在は偶然であるとの主義は科学者とか教授とかに一般に賛成しているらしいです。  証拠はあるでしょうか？宇宙全体で生命の希少性は極みに高いです。生命の存在は軌跡にしか考えられません。訳の証拠は下記で表そうとします。

僕の論理を理解できるようにまず基本のことを定義して説明します。

確率かくりつ

国語辞典で調べたらこの定義があります:

かく‐りつ【確率】
《probability》
ある事象の起こる可能性の度合い。公算。蓋然率(がいぜんりつ)。「―が高い」

例:

袋にはボールの10個があるとしましょう。 ボールは1から10までの番号が付いています。 見ずに1号のボールを出す確率は何でしょうか?
 正解は10分の1ですね、１割、１/１０、１０％です。

 方式は

確率　＝	事象の起こる場合の数
	起こりうるすべての場合の数

順列じゅんれつ

国語辞典の定義:

じゅん‐れつ【順列】
《permutation》
１ 順序に従って並べること。
２ 数学で、n個のものからr個を取り出し、順序を決めて1列に並べたもの。

例:

ボールの3個があるとしましょう。
ボールは1から3までの番号が付いています。
ボールを並べるとしたら識別できる並び方の数は何でしょうか?

1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1

6つの識別できる並び方がありいます。
4号のボールを追加したら並び方はいくつありますか?

1,2,3,4
1,2,4,3
1,3,2,4
1,3,4,2
1,4,2,3
1,4,3,2
2,1,3,4
2,1,4,3
2,3,1,4
2,3,4,1
2,4,1,3
2,4,3,1
3,1,2,4
3,1,4,2
3,2,1,4
3,2,4,1
3,4,1,2
3,4,2,1
4,1,2,3
4,1,3,2
4,2,1,3
4,2,3,1
4,3,1,2
4,3,2,1

24つの並び方があります。
5号ボールを追加したら同でしょうか？
正解は１２０です。（１ｘ２ｘ３ｘ４ｘ５＝１２０）
方式:
順列の合計=ｎｘ（ｎ－１）ｘ（ｎ－２）ｘ（ｎ－３）ｘ。。。ｘ１
この方式を略して（ｎ！）と書きます。（ｎの階乗かいじょう）と呼ばれています。

このサイトを参照:
http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

確率と順列をあわせたら。。。

順列と確率をあわせて面白い質問を検討できます。

トランプカードのデッキには52枚のカードが入っています。
デッキを混ぜても順序の並びが偶然に出る確率は何でしょうか?

まず、順列の数を計算します。
52枚のカードだから52!（52の階乗）の並び方があります。
52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
科学的記数法で8.0658ｘ10⁶⁷です。
52枚があるのにすごい数ですね！

一億の並び方を１秒の割合で試すことにしたら全部の並び方を試すのに時間どのくらいかかるでしょうか？2.55590ｘ10⁵²年がかかります！

では、確率は？
1つの正しい順序並び方しかないので確率は1.0 / 8.0658ｘ10⁶⁷ = 0.1240ｘ10^-67です。偶然に正しい並び方を選ぶのが無理にしか考えられません。言い換えれば可能性は極みに低いですね。

生命は偶然ではない証拠:

生命にはDNAが必要です。DNAは動物の図面みたいです。
DNAには塩基対が特別な並べ方があって正しくない並び方じゃない限り、動物の生命はできないと考えられます。この話は塩基配列についてです。wikipedia参照:http://ja.wikipedia.org/wiki/DNA

英語版のwikipediaからヒトゲノムのページではこの表があります。(私は合計を追加しました）
http://en.wikipedia.org/wiki/Human_genome
http://ja.wikipedia.org/wiki/%E3%83%92%E3%83%88%E3%82%B2%E3%83%8E%E3%83%A0

染色体	長さ (mm)	塩基対	変異	確認たタンパク	推定タンパク	偽遺伝子	miRNA	rRNA	snRNA	snoRNA	他ncRNA
1	85	249,250,621	4,401,091	2,012	31	1,130	134	66	221	145	106
2	83	243,199,373	4,607,702	1,203	50	948	115	40	161	117	93
3	67	198,022,430	3,894,345	1,040	25	719	99	29	138	87	77
4	65	191,154,276	3,673,892	718	39	698	92	24	120	56	71
5	62	180,915,260	3,436,667	849	24	676	83	25	106	61	68
6	58	171,115,067	3,360,890	1,002	39	731	81	26	111	73	67
7	54	159,138,663	3,045,992	866	34	803	90	24	90	76	70
8	50	146,364,022	2,890,692	659	39	568	80	28	86	52	42
9	48	141,213,431	2,581,827	785	15	714	69	19	66	51	55
10	46	135,534,747	2,609,802	745	18	500	64	32	87	56	56
11	46	135,006,516	2,607,254	1,258	48	775	63	24	74	76	53
12	45	133,851,895	2,482,194	1,003	47	582	72	27	106	62	69
13	39	115,169,878	1,814,242	318	8	323	42	16	45	34	36
14	36	107,349,540	1,712,799	601	50	472	92	10	65	97	46
15	35	102,531,392	1,577,346	562	43	473	78	13	63	136	39
16	31	90,354,753	1,747,136	805	65	429	52	32	53	58	34
17	28	81,195,210	1,491,841	1,158	44	300	61	15	80	71	46
18	27	78,077,248	1,448,602	268	20	59	32	13	51	36	25
19	20	59,128,983	1,171,356	1,399	26	181	110	13	29	31	15
20	21	63,025,520	1,206,753	533	13	213	57	15	46	37	34
21	16	48,129,895	787,784	225	8	150	16	5	21	19	8
22	17	51,304,566	745,778	431	21	308	31	5	23	23	23
X	53	155,270,560	2,174,952	815	23	780	128	22	85	64	52
Y	20	59,373,566	286,812	45	8	327	15	7	17	3	2
mtDNA	0.0054	16,569	929	13	0	0	0	2	0	0	22
合計	699	2,057,493,819	37,096,106	11196	369	8765	1090	337	1291	864	790

ヒトゲノムは2,057,493,819の塩基対があります。その中37,096,106の有効の変異があります。
全部の塩基対の中の有効の変異の以外は無効としましょう。
有効変異を偶然に選ぶのにどの確率がありますか？

(2057493819 - 37096106)! = 2020397713! = 1.2622ｘ10^{17,923,235,794}

ですから偶然に有効なヒトゲノムを選ぶの確率は1.0 / 1.2622ｘ10^{17,923,235,794} =  7.9227ｘ10^{-17,923,235,794}<です。考えられないくらい少ないです。一番小さい生き物にしても約5500万の塩基対があるので確率は1 / 4.8615ｘ10^401,833,755です。考えられない程、少ないです。

ですから「生命は偶然に生じた」と言う主義は笑うべきでしょう。

宇宙全体に数えられないくらい惑星があります。しかしそれぞれの惑星があってもDNAの構成自体が偶然に組み合わせることがあるとしても完璧な環境、栄養とか熱とかがあるとしても（可能性が少なくとも）それぞれのことが丁度いいにしてもDNAの塩基配列が完璧であるにしても（上記の話参照）生命が生じると考えるでしょうか?

つまり神様がなければ生命が生じる可能性は０にしか考えられないのです。