Trinomial Distribution Connection
The numbers of the Tetrahedron can also be found in the Trinomial Distribution. This is a discrete probability distribution used to determine the chance some combination of events occurs given three possible outcomes−the number of ways the events could occur is multiplied by the probabilities that they would occur. The formula for the Trinomial Distribution is:
where x, y, z are the number of times each of the three outcomes does occur; n is the number of trials and equals the sum of x+y+z; and PA, PB, PC are the probabilities that each of the three events could occur.
For example, in a three-way election, the candidates got these votes: A, 16%; B, 30%; C, 54%. What is the chance that a randomly-selected four-person focus group would contain the following voters: 1 for A, 1 for B, 2 for C? The answer is:
× = 12 × 0.0140 = 17%
The number 12 is the coefficient of this probability and it is number of combinations that can fill this "112" focus group. There are 15 different arrangements of four-person focus groups that can be selected. Expressions for all 15 of these coefficients are:
The numerator of these fractions (above the line) is the same for all expressions. It is the sample size−a four-person group−and indicates that the coefficients of these arrangements can be found on Layer 4 of the Tetrahedron. The three numbers of the denominator (below the line) are the number of the focus group members that voted for A, B, C, respectively.
Shorthand is normally used to express combinatorial functions in the following "choose" format (which is read as "4 choose 4, 0, 0", etc.).
But the value of these expression is still equal to the coefficients of the 4th Layer of the Tetrahedron. And they can be generalized to any Layer by changing the sample size (n).
This notation makes an easy way to express the sum of all the coefficients of Layer n:
Read more about this topic: Pascal's Pyramid
Famous quotes containing the words distribution and/or connection:
“There is the illusion of time, which is very deep; who has disposed of it? Mor come to the conviction that what seems the succession of thought is only the distribution of wholes into causal series.”
—Ralph Waldo Emerson (1803–1882)
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—Friedrich Nietzsche (1844–1900)