The figure below depicts a model much like the sane-demented model with which we introduced matrices. However, this model involves three categories instead of two. The three categories are sane, borderline and demented.

As depicted in the figure, we have the following proportional changes in the three categories:

• .10 (or 10%) of the sane will become demented and .15 will become borderline during a transition period,
• .15 of the demented will become sane and .15 will become borderline during a transition period
• .10 of the borderline will become demented and .20 will become sane during a transition period.

We assume that the initial population consists of 500 sane, 300 borderline and 200 demented.

The figure below shows how the numbers change in this transition.

We see that

• the sane population loses 50 + 75 and gains 40 + 45 for a net loss of 40,
• the demented population loses 45 + 45 and gains 50 + 20 for net loss of 20, and
• the borderline population loses 40 + 20 and gains 75 + 45 for net gain of 60.

This results in a sane population of 500 - 40 = 460, a demented population of 300 - 20 = 280 and a borderline population of 200 + 60 = 260, as indicated by the numbers at the bottom of the figure.

• The process could be repeated with these numbers, then repeated again with the resulting population numbers, etc., for as long as we wish.

The matrix multiplication below has the same effect as the calculation process above.

• When the first line of the matrix is multiplied by the column vector on the left, the result includes the proportions .20 of the 200 borderline and .15 of the 300 demented, who all become sane, plus the .75 of the sane who remain sane. The resulting 460 represents the number of sane after the transition.
• When the second line of the matrix multiplied by the column vector on the left, the result includes the proportions .15 of the sane who become borderline, the .70 of the borderline who remain borderline and the .15 of the demented who become borderline. The resulting 260 represents number of borderline after the transition.
• A similar interpretation of the last line shows us that the number (280) of the demented after the transition is obtained using the same numbers as in the calculation above.

Video File #01

The up, down, same behavior of the stockmarket can be modeled by a diagram similar to that used with the preceding population model.

This model tells us that

• if the stocks are up today, there is a .5 probability that they will be down tomorrow and a .25 probability that they will be unchanged tomorrow, and an implied probability of .25 that they will be up tomorrow;
• if the stocks are down today, there is a .25 probability that they will be up tomorrow and a .25 probability that they will be unchanged tomorrow, which implies a probability of .5 that the stocks will be down again tomorrow;
• if the stocks are the same today, there is a .25 probability that they will be up tomorrow and .25 probability that they will be down tomorrow, which implies a .5 probability that they will be the same tomorrow.

The first line of the matrix below tells us that

• to get the probability of stocks being up tomorrow, we must add the .25 U to the .25 D and the .25 S that follow from the rule;
• to get the probability of stocks being down tomorrow, we must add the .5 U to the .5 D and the .25 S that follow from the rule,
• to get the probability of stocks being the same tomorrow we must add the .25 U to the .5 D and the .25 S that follow from the rule.

We note that the matrix is a stochastic matrix, with each column adding up to 1.

• The first column tells us that we split up the U into .25 U (U staying U), the .5 U (.5 U becoming D) and .25 U (U becoming S).
• The second column tells us how we split up the D (.25 D becoming U, .5 D staying D and .25 D becoming S).
• The third column tells us how S is split up.

Each column tells us how one of the quantities is split up; thus the proportions represent the entire quantity and must add to 1.

In the figure below, U, D and S are replaced by p₁, p₂ and p₃. Nothing else in the matrix multiplication is changed.

• We now interpret U, D and S as probabilities p₁, p₂ and p₃ that stocks are up, down or the same today.
• For example if we somehow determine that the probabilities of stocks being up and being down at the end of today are both 1/2, and that there is no probability that they will be the same, then the probability of stocks being up tomorrow will be

• p₁⁺ = 1/4 p₁ + 1/4 p₂ + 1/4 p₃ = 1/4 * 1/2 + 1/4 * 1/2 + 1.4 * 0 = 1/4.

• Similar calculations can be used to find p₂⁺ and p₃⁺ for these assumed probabilities for today.

Video File #02