Natural Selection and Evolution

How Fast Does Evolution Accumulate Tries?

Before investigating how natural selection limits the number of tries, consider a simple example.

Suppose that a scientist is given three dice and told to roll them until he throws triple fives. The odds that he will throw triple fives on the first roll are 1 in 216 (the dice have 6 x 6 x 6 = 216 possible outcomes, and only one is triple fives). What are the odds when the scientist throws the three dice twice? Many readers may think that the odds double. But this is only an approximation, and the approximation is only accurate if the odds are poor. The equation required to calculate the odds is as follows: odds of triple fives = 1 - (215/216) ^{number of rolls} . So with one roll the odds are 1 - (215/216) ¹ = 1/216 or 1 in 216. The odds with 2 rolls are 1-(215/216)²= 1/108.25 or 1 time in 108.25 tries. Notice that the odds did not quite double

Rolls   Probability             Odds

1          0.46%                1 in 216

2          0.92%                1 in 108.25

4          1.84%                1 in 54.4

8          3.65%                1 in 27.4

16        7.2%                  1 in 14

32         13.8%               1 in 7.2

64         25.7%               1 in 3.9

128        44.8%              1 in 2.2

256        69.5%              1 in 1.4

512        90.7%              1 in 1.1

1024      99.1%              1 in 1.01

Figure 15.4 uses a bar to represent the probability of rolling triple fives. The numbers along the bottom represent the number of tries. When the bars are short, each successive bar is almost twice as high as its predecessor. Once the probability is greater than ten percent, doubling the tries no longer doubles the probability. The probability for success will never be equal to 100%, but after 1024 tries, it is very close.

Figure 15.4: Probability of Rolling Triple Fives

With quite a bit of mathematical manipulation, figure 15.4 can be converted into figure 15.5.

Figure 15.5: A Growing Tree Helps the Scientist Climb the Wall

In figure 15.5, each bit represents one foot. Because rolling triple fives corresponds to a 1 in 216 chance, the initial height of the wall in figure 15.5 is 7.75 bits (information = 3.32xlog(216/1) = 7.75 bits) or 7.75 feet.

       After two rolls, the height of the wall is given as follows: information = 3.32 x log(108.25/1) = 6.75 bits. Rather than shrink the wall, which is hard to draw, figure 15.5 shows the scientists standing on a tree. The height of the tree is the initial height of the wall minus the new height of the wall. Thus, after two rolls the tree is 7.75- 6.75 = 1 foot high.

       After 16 rolls, the odds improve to 1 in 14. Thus, the new height of the wall is equal to 3.32 x log (14/1) = 3.8 bits or 3.8 feet. To compensate the tree must be 4 feet tall (7.75 - 3.8 = 3.95).

        The scientist is standing on the tree that corresponds to 16 rolls. He only has a 1 in 14 chance of climbing the wall. After 1024 rolls, there is almost no chance that he will not be able to climb over the wall.

        To relate this example to evolution, each roll of the dice corresponds to a try, and each try corresponds to a reproductive event. So how fast the tree grows depends on reproductive rates. Animals that have large populations accumulate many more tries than those with small populations. Animals that reproduce slowly like elephants will accumulate fewer tries than animals that reproduce quickly like rabbits. Since it takes time to accumulate tries, the number of tries can easily be converted into years. If the scientist rolls the dice once a year, then the x-axis in figure 15.5 can be written in years, and it will take the tree that the scientist is standing on 16 years to grow.

   This technique is not limited to biological evolution. It can also be used to model chemical evolution. Every time a chemical polymerizes in the primordial soup, the odds of creating a self-replicating molecule roughly double. Furthermore, every planet in the universe may have its own primordial soup. Both of these factors will significantly improve the odds of a self-replicating molecule evolving somewhere in the universe.

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