APPENDIX 1: STANDARD PROBABILITY THEORY

Is it hard to learn
probability theory? No, not at all. It involves
nothing more than figuring the odds at a dice game, or a game of cards. For example,
the probability of rolling three ‘6’s in sequence using standard gaming dice is
*not* computed by *adding* the difficulty level of rolling one ‘6’ to
that of rolling the others. Doing it that way would yield 6 + 6 + 6, or 1
chance in 18, which is incorrect. The correct method is to multiply the
probability of rolling a given value on one die times the probability of
rolling a given value on the next, 6 X 6 X 6. This is because there are 36
alternative ways of rolling two dice in sequence, and 216 ways of rolling 3
dice *in sequence*, assuming that
each die is labeled with an identifying name and rolled separately, such as die
‘A,’ die ‘B,’ and die ‘C.’

If we label the dice and roll them separately the chances of rolling A6, followed by B6 followed by C6 are only 1 chance in 216. If we don’t label the dice and don’t monitor which is rolled first and only care about the values, then there will be 3 chances out of 216 to get three dice with a value of 6 for each of the three dice, regardless of the order in which they are rolled.

Consider the following tables which list the alternative possibilities from rolling two or three dice in sequence that have been labeled ‘A,’ ‘B,’ and ‘C’ for identification of each separate die.

Table 1—36 Alternative Outcomes for Rolling Two Dice

A1 |
B1 |

A1 |
B2 |

A1 |
B3 |

A1 |
B4 |

A1 |
B5 |

A1 |
B6 |

A2 |
B1 |

A2 |
B2 |

A2 |
B3 |

A2 |
B4 |

A2 |
B5 |

A2 |
B6 |

A3 |
B1 |

A3 |
B2 |

A3 |
B3 |

A3 |
B4 |

A3 |
B5 |

A3 |
B6 |

A4 |
B1 |

A4 |
B2 |

A4 |
B3 |

A4 |
B4 |

A4 |
B5 |

A4 |
B6 |

A5 |
B1 |

A5 |
B2 |

A5 |
B3 |

A5 |
B4 |

A5 |
B5 |

A5 |
B6 |

A6 |
B1 |

A6 |
B2 |

A6 |
B3 |

A6 |
B4 |

A6 |
B5 |

A6 |
B6 |

Table
2—35 New Options Added for __Each__ Value of a Third Die

A1 |
B1 |
C1 |

A1 |
B2 |
C1 |

A1 |
B3 |
C1 |

A1 |
B4 |
C1 |

A1 |
B5 |
C1 |

A1 |
B6 |
C1 |

A2 |
B1 |
C1 |

A2 |
B2 |
C1 |

A2 |
B3 |
C1 |

A2 |
B4 |
C1 |

A2 |
B5 |
C1 |

A2 |
B6 |
C1 |

A3 |
B1 |
C1 |

A3 |
B2 |
C1 |

A3 |
B3 |
C1 |

A3 |
B4 |
C1 |

A3 |
B5 |
C1 |

A3 |
B6 |
C1 |

A4 |
B1 |
C1 |

A4 |
B2 |
C1 |

A4 |
B3 |
C1 |

A4 |
B4 |
C1 |

A4 |
B5 |
C1 |

A4 |
B6 |
C1 |

A5 |
B1 |
C1 |

A5 |
B2 |
C1 |

A5 |
B3 |
C1 |

A5 |
B4 |
C1 |

A5 |
B5 |
C1 |

A5 |
B6 |
C1 |

A6 |
B1 |
C1 |

A6 |
B2 |
C1 |

A6 |
B3 |
C1 |

A6 |
B4 |
C1 |

A6 |
B5 |
C1 |

A6 |
B6 |
C1 |

In Table 1 we see that there are 36 alternative outcomes when rolling two dice that are labeled for identification—not 12. When we roll a third die, many more alternative outcomes are added, not just 6. Table 2 shows the 36 alternatives for the set of three dice available that have the value 1 on the third die. Another 36 are created for each of the remaining values, 2-5, for a total of 216 alternatives for the set of three dice.

In the “real world”
the importance of order of events is a bit easier to see than in games because,
in the events of physics, chemistry, and biology, the order of occurrence in
time is more relevant to the result. Dice are physically identical for purposes
of gaming but rarely are two physical events fully identical.

Consider the
following two simplistic event sequences: buy meat, cook meat, eat meat; make
bomb, light bomb, run. If you eat the meat before you buy it, the store manager
will react differently than if you buy it first. If you eat it before you cook
it, *you* will react differently. Lighting the fuse before the bomb is
built will obviously not produce a bang, and so on. In gaming, rolling a ‘6’
before a ‘3” makes no difference, but in real world events sequence nearly
always matters.

Most natural events of physics, chemistry, and biology are sensitive to sequence in the same way. Forming air-breathing creatures before the plants of the Earth have produced any oxygen to breath obviously has a different result than making the oxygen first. A germ mutating before it infects its host has a different effect than its mutating after it leaves the host, etc. The difference can be life and death. The order in which chemicals are mixed is similarly important. With the joining of the first two chemicals new compounds are made; energy is released; gases are formed and transformed, etc. When the third chemical is added it will find conditions different in important ways than if it had been added to the first chemical directly. The effect on the environment will also be different depending upon sequence. Mixing two caustic chemicals directly and somewhat later diluting that mixture heavily with water as a second step results in poisoned air. However, initially diluting the first chemical with water and somewhat later adding the second chemical does not produce noxious gas.

Calling probability theory “game theory” can sometimes be a little misleading when the real world is what one is applying the theory to. Rolling craps in Vegas, one gives no thought to which die comes out first, but, in nature the sequence of events matters. Nature is a different kind of game that almost never offers two events that have exactly the same physical characteristics. But even where event 1 and event 3 are identical it can make all the difference that event 2 has occurred between them.

For this reason, when using dice to model the total alternatives available for probability theory computations, each die is labeled and tracked separately. Having a ‘6’ on die ‘A’ as far as Mother Nature is concerned can be a totally different situation than having a ‘6’ on die ‘C.’ In the physical and chemical interactions of nature we see that there really are 216 relevant possible outcomes for three “dice.” Having a ‘3’, a ‘5’, and a ‘6’ is not the same as having a ‘6’, a ‘5’, and a ‘3’.

There are some situations in nature where sequence is not important, but when building proteins from amino acids, and building the DNA sequences that direct which amino acid will be built, sequence is relevant. True, nature seems to have built those proteins much more quickly than our probability figures would indicate could be done randomly. But that doesn’t tell us our probability computations are wrong; math is math. It tells us that nature didn’t use an accidental process. Probability theory only applies to accidental or random event processes. A process guided by intelligent design or purpose is much more efficient than a random/accidental process.