Introduction to the Mathematics of Evolution

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

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 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 and a '1' is achieved 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 page is what you calculate for 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.

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 the "first living cell" would be 10^‑79.

Thus, even the changes of a "first living cell" (which is only the very, very beginning of evolution), are virtually impossible, even at 5,000 attempts every second 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 10^1,400 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.

Consecutive Lotteries

Now suppose there was a second lottery (all lotteries in this section are assumed to have a probability of 10^‑100). However, according to the rules, you could not buy a ticket in this second lottery until after you won the first lottery. This could be called a "consecutive probability" or "consecutive lottery" because one probability has to occur before you can even be eligible to buy tickets in the second lottery.

It is unlikely you would ever be eligible to buy tickets in the second lottery because it would be virtually impossible you would win the first lottery. But if you did win the first lottery, the chances are you would not have had the time left to buy very many tickets in the second lottery. Thus the chances of winning the second lottery (after winning the first lottery) are ludicrous. Not impossible, but ludicrous.

Now suppose there were 150 "consecutive lotteries?" For example, you would have to win the first lottery before you were allowed to buy a ticket in the second lottery. You would have to win the first lottery and then the second lottery, before you could even buy a ticket in the third lottery. You would have to win the first lottery, then the second lottery, then the third lottery, before you could even buy a ticket in the fourth lottery. And so on.

What are your chances of winning all 150 lotteries within 5 billion years? Absolute, pure insanity; that is your probability. Is it impossible? In a real world, yes it would be impossible. In fact, we could say that in a real world it would be impossible to win 5 consecutive lotteries under the above rules.

As will be shown in a future chapter, the probability of the theory of evolution being true is worse than winning 150 consecutive lotteries. In fact, the probability is much, much worse.

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.