Introduction to the Mathematics of Evolution

 

Chapter 14

 

Understanding Big and Little Numbers

 

 

"Philosophy is a game with objectives and no rules.  Mathematics is a game with rules and no objectives."

Ian Ellis

 

 

Understanding Really Big Numbers

 

One of the hardest things for human beings to do is comprehend the difference between a number like 900 versus a number like 10900.

 

Both numbers have the symbols '900' in them.  Thus, when someone sees a number like 10900 they naturally think of the number 900, and don't see much difference between 900 and 10900.

 

The number 900 is just that, a number which any middle-school student can count to in a matter of a few minutes.  If a person counted to 900, one number per second, they would count to 900 in 15 minutes.

 

If we paid $900 for a television set, we would see our bank account drop by $900.

 

But how long would it take us to count to 10900?

 

First of all, let us look at the number 10900 written longhand.

 

The number 10900 is a '1' followed by 900 zeros.  This is what it looks like:

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000

 

Each consecutive zero in this number represents a number which is 10 times larger than the number before it.  For example, 100 is 10 times larger than 10.  1,000 is 10 times larger than 100.  10,000 is 10 times larger than 1,000.  And so on.  Thus, we are essentially multiplying 10, by itself, 900 times.

 

If we were to write out a much, much smaller number (the number of atoms in the known Universe), we would write it out:

100,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000

 

The number of atoms in 10 Universes (1081) would be written out:

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000

 

The number 10900 is the number of atoms in:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000 Universes!!

 

The above number is 10(900‑80) or 10820.  The above number is how many Universes there would have to be in order to be able to count 10900 different atoms.

 

While we could easily count to 900 in a few minutes, we could not count to 1080 during our entire lifetimes, much less count to 10900.

 

It is absolutely critical for the reader to comprehend the difference between the number 900 and the number 10900.  The number 10900 is a number which is humanly incomprehensible.  It represents a huge, huge, huge number and represents how many zeros follow the initial 1.

 

The point to this discussion is when you see a number like 10900; do not think of the number 900; rather think of a 1 followed by 900 zeros.  Also think about the fact that it represents the number of atoms in 10820 Universes!!

 

 

What Constitutes an "Impossible" Event?

 

Now we will talk about really small numbers.

 

In this book, a probability of 10‑100; meaning a situation where only 1 out of 10100 chances or attempts would be considered a "success"; is defined to be "impossible."

 

Obviously, a probability of 10‑500, or any other number less than 10‑100, would also be considered "impossible."

 

While technically, nothing is impossible, this level of probability is so rare, for all practical purposes, a probability of this magnitude will never happen during the age of our planet.

 

As mentioned above, there are about 1080 atoms in our Universe.  That is a '1' followed by 80 zeros.

 

How much smaller is 10‑100 than 10‑80?  The answer is 10(‑100‑80) or 10‑20.  Thus, picking the correct single atom from among 10100 atoms is much harder than picking the single correct atom in our Universe.

 

In fact, the probability of 10‑100 is equivalent to picking a single, correct atom, from among all the atoms in 1020 or:

100,000,000,000,000,000,000 different Universes!!  The atoms in our Universe would only constitute a very, very, very minute percentage of the 10100 atoms in this many Universes.

 

While as mentioned above, 10‑100 is technically not an impossible probability, let us study some examples of just how big it really is.

 

Suppose there was a lottery in which a 10-sided dice was rolled 100 times.  In order to win this lottery you had to roll a '1' for all 100 rolls.  In other words, you had to roll a '1' for 100 consecutive rolls, including the first roll, the second roll, the third roll, the fourth roll, etc.

 

It sounds simple doesn't it?  It turns out that rolling a '1' for 100 consecutive times is equivalent to picking the correct, single atom from among 1020 Universes, where each Universe has 1080 atoms!!

 

Each "ticket" in this lottery represents your attempt to roll the dice 100 consecutive times where the roll is a '1' in every attempt.  If you rolled something other than a '1' your ticket immediately fails and you quit rolling the dice.  Thus, if you roll a '5' on the first roll, there is no need to make any more rolls, your ticket has failed.

 

As another example, suppose for one "ticket" you rolled:

First roll:  a '1'

Second roll: a '1'

Third roll: a '4'

 

You would stop after the third roll since the third roll was not a '1'.  This "ticket" failed also.

 

In a computer simulation of rolling dice, 50 billion attempts were made to roll 100 '1's in a row.  Here are the results of this computer simulation:

 

 

Table: Maximum number of times a '1' was rolled at the beginning:

 

Note: The first item in the table means a '1' was not rolled in the first attempt.  The second item in the table means a '1' was rolled on the first attempt, but not the second attempt.  And so on.

 

 Rolled a '1' [0] consecutive times:  Count = 44,999,935,077

 Rolled a '1' [1] consecutive times:  Count = 4,500,063,675

 Rolled a '1' [2] consecutive times:  Count = 449,993,542

 Rolled a '1' [3] consecutive times:  Count = 45,006,419

 Rolled a '1' [4] consecutive times:  Count = 4,500,592

 Rolled a '1' [5] consecutive times:  Count = 450,545

 Rolled a '1' [6] consecutive times:  Count = 44,967

 Rolled a '1' [7] consecutive times:  Count = 4,682

 Rolled a '1' [8] consecutive times:  Count = 454

 Rolled a '1' [9] consecutive times:  Count = 43

 Rolled a '1' [10] consecutive times: Count = 4  [max]

 

In other words, in 50 billion attempts, the closest to 100 consecutive '1's in a row was 10 in a row.  And this only happened 4 times out of 50 billion attempts.

 

Most people would think that it would be easy to roll 20 '1's in a row.  But in 50 billion attempts, the most number of '1's in a row was 10.

 

Suppose you were given this offer: "If you invest your life's savings in this lottery (the lottery to roll 100 '1's in a row), you will be given 5,000 tickets (i.e. 5,000 attempts to win the lottery), for every second in a 5 billion year period."

 

In other words, we will assume this earth is 5 billion years old and you are given 5,000 tickets (i.e. attempts) every second; 24 hours a day, 365.25 days a year, for the entire time the earth has existed!!

 

Assuming your life's saving were $1,000,000, would you invest your life's savings in this lottery?  Answer that question before reading on.

 

Let us see your odds of winning.  We will assume you will be able to buy, at 5,000 tickets a second:

1,000,000,000,000,000,000,000 tickets in 5 billion years (actually you would be able to buy slightly less than that).  This is 1021.

 

This is your last chance; would you spend your life's savings on these 1021 tickets?

 

To calculate your odds of winning, we do this simple calculation: 10(21‑100) = 10‑79.  In other words, your odds of winning this lottery, even with 1021 tickets, is only 1 chance in 1079.  This is about the same as picking the single correct atom from all the atoms in our Universe.

 

But let's suppose you didn't know the simple way to calculate your odds.

 

The next chart shows how to calculate your odds.

 

 

Chart A

 

Based on 1,000,000,000,000,000,000,000 Tickets (1021)

 

"0 ct" means the first roll was not a '1'

"1 ct" means the first roll was a '1', but not the second roll

"2 ct" means the first two rolls were a '1', but not the third roll, etc.

Only the "100 ct" line below is a winner.

 

Symbol  Probability       Predicted # of Times Rolled

 0 ct   .9                           9 x 1020

 1 ct   .09                          9 x 1019

 2 ct   .009                         9 x 1018

 3 ct   .0009                        9 x 1017

 4 ct   .00009                       9 x 1016

 5 ct   .000009                      9 x 1015

 6 ct   .0000009                     9 x 1014

 7 ct   .00000009                    9 x 1013

 8 ct   .000000009                   9 x 1012

 9 ct   .0000000009                  9 x 1011

10 ct   .00000000009                 9 x 1010

11 ct   .000000000009                9 x 109

12 ct   .0000000000009               9 x 108

13 ct   .00000000000009              9 x 107

14 ct   .000000000000009             9 x 106

15 ct   .0000000000000009            9 x 105

16 ct   .00000000000000009           9 x 104

17 ct   .000000000000000009          9 x 103

18 ct   .0000000000000000009         9 x 102

19 ct   .00000000000000000009        9 x 101

20 ct   .000000000000000000009       9 x 100

21 ct   .0000000000000000000009      9 x 10‑1

22 ct   .00000000000000000000009     9 x 10‑2

23 ct   .000000000000000000000009    9 x 10‑3

24 ct   .0000000000000000000000009   9 x 10‑4

25 ct   .00000000000000000000000009  9 x 10‑5

...

98 ct                                                   9 x 10-78

99 ct                                                  9 x 10-79

100 ct (the only winner)                       9 x 10‑80 (approx 10-79)

 

Even though you own 1021 tickets, which is a huge number of tickets, your chance of winning is only 10‑79.  As mentioned above, this just happens to be about the same probability as picking the correct, single atom, from among all the atoms in our Universe.

 

Thus, even though you get 5,000 tickets, every second, every day, every year for 5 billion years, your chance of winning this lottery is about the same as picking the single correct atom from among all the atoms in our Universe.

 

Would you spend you life's saving to enter this lottery?  Well, would you spend your life's savings on picking the correct, single atom, from among all the atoms in our Universe?  It is effectively the same question.

 

If you only bought one ticket, your chances would be the same as picking the single, correct atom, from among all the atoms in 1020 Universes!!  Your chances would be 10‑100.

 

Hopefully you would not buy a single ticket in this lottery.  You would save a lot of time and gasoline by simply flushing your dollar bills down the toilet.

 

The point to this exercise is that an event which has a probability of 10‑100 is an event which is very, very unlikely to happen, a single time, in the age of our earth!!  This is true even if there are 5,000 events (i.e. 5,000 tickets), every second, for the entire age of our earth.

 

 

The "First Living Cell"

 

Now let us assume the "first living cell" of evolution had 900,000 nucleotides.  How many permutations of 900,000 nucleotides are there?  The answer is 4900,000.

 

How much bigger is 4900,000 than 4150, and remember that the number 4150 is bigger than the number of atoms in our Universe?

 

Try to calculate it before reading on.

 

If you said 6,000 times bigger, you would be wrong.  The correct answer is 4899,850 times bigger!!!

 

Remember, when you are dividing exponents, which have a common base, you subtract their exponents, you do not divide their exponents.  Thus, 4900,000 divided by 4150 is equal to 4(900,000‑150) = 4899,850.

 

And this is just the "first living cell."  Human DNA has 3,000,000,000 pairs of nucleotides!!  There are 43,000,000,000 unique permutations of 3 billion nucleotides.

 

This is just an introduction to the subject of permutations of nucleotides.

 

Now let apply the "first living cell" permutations to our probability which is defined to be "impossible."

 

For example, suppose someone calculated the probability of the "first living cell" to be 10‑100 (actually the probability of the "first living cell" is much, much lower than that).  Furthermore, suppose scientists were able to create 5,000 attempted "first living cells" every second, for 5 billion years.  Their chance of creating a single "first living cell" would be 10‑79.

 

Thus, even the chance of a "first living cell" (which is only the very, very beginning of evolution), is virtually impossible, even at 5,000 attempts every second, 24 hours a day, for the age of our earth.  And in the real world there would probably only be a few hundred attempts every century (and that is very generous to the theory of evolution).

 

The real probability of the "first living cell" is not 10‑100, but it is about 10‑1,500, which is 101,400 times smaller than the impossible probability of 10‑100!!

 

Without the "first living cell," there is no evolution.

 

The "impossible" probability of 10‑100 effectively takes into account a large number of events which might be "winners," namely 5,000 possible events every second.  But even with a large number of attempts to "win the lottery," a person is left with essentially an impossible probability.

 

 

Comment

 

The chapters on mathematics have covered a lot of concepts in a short amount of space.  If you do not feel comfortable with these concepts, you would be advised to read these chapters again and even get some help from a friend or relative.