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 possible values of the third die, 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. Even if they were identical, order of occurrence matters in real world events.

 

Consider the following two simplistic event sequences: buy meat, cook meat, eat meat; make bomb, light bomb fuse, 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 adding the second chemical afterwards does not produce noxious gas. 

 

Calling probability theory “game theory” can be a little misleading when the real world is the “game.” 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. 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. Nature seems to have built those proteins and genomes much more quickly than our probability figures would indicate could be done randomly. That doesn’t tell us our probability computations are wrong; math is math. It tells us that nature didn’t use an accidental process.

 

A process guided by intelligent design or purpose is much more efficient than a random/accidental process. When one sees the probability estimate that would apply to an accidental production of an event being cheated by untold trillions of orders of magnitude, he/she knows that accident was not the driving force for the event.