is Caucd increasing the confidence interval. You would not raise the time to 2 hours, because the likelihood of such a delay would be too remote. Estimates imply an allowable variation, all of which is considered normal.

Probability is the measuring of the uncertainty surrounding an average value- Probabilities are measured in percent of likelihood. For exanple, if M numbers fran a total of N are experted to fall within a specific range, the probability P of any one number satisfjing the criteria is

When making a trade, or forecasting prices, we can only talk in terms of probabilities or ranges. We expert prices to rise 30 to 40 points, or we have a chance of a $400 profit fran a trade. Nothing is certain, but a high probability of success is very atfractive.

Laws of Probability

Two basic principles in probability are easily explained by using examples with plajing cards. In a deck of 52 cards, there are 4 suits of 13 cards each. The probattlity of drawing a specific card on any one tum is 1/52. Similarly, the chances of drawing a particular suit or card number are 1/4 and 1/13, respertively. TTie probability of any one of these three possibilities occurring is the sum of their individual probabilities. This is known as the law of addition. The probability of success in choosing a numbered card, suit, or specific card is

Another basic principle, the law of multiplications, states that the probability of two occurrences happening simultaneously or in succession is equal to the produrt of their separate probabilities. The likelihood of drawing a three and a club from the same deck in two consecutive tums (replacing the card after each draw) or of drawing the same cards from two deds simultaneously is

Joint and Marginal Probability

Price movement is not as clearly defined as a deck of cards. There is often a relationship between successive events. For exanple, over two consecutive daj-s, prices must have one of the following sequences or joint events., (up, up), (down, down), (up, down), (down, up), with the joint probabilities of .40, .10, 35, and .15, respertively in this , there is the greatest expedation that prices will rise. The marginal probability of a price rise on the firet day is shown in Table 2-2. Thus there is a 75>o chance of higher prices on the first day and a 55>o chance of higher prices on the second day.

Contingent Probability

What is the probability of an outcome contingent on the result of a prior event? In the example of joint probability, this might be the chance of a price increase on the second dsy

when prices declined on the first day. The notation for this situation (the probability of A conditioned on B) is

J.Unt prt>bability «,f <cl«>w

/-it-ify Day 2 I dow«i Uay > =

Tl¹ F><-obat>llliy off Ithf pries •ncrease on I >:>y I cir price increase on I >;i}- ,2 is f (ellli«rO = » 1 > -*- >-<up Day 2> - up Day 1 ami up Day 2>

= .TS -1- .SS - iO

Markov Chains

If we believe that todays price movement is based in some part on what happened yesterday, we have a situation called conditional probability. This can be expressed as a Markov process, or Markov chain. The results, or outcomes, of a Markov chain express the probability of a state or condition occurring. For example, the possibility of a clear, cloudy, or rainy day tomorrow might be related to todays weather.

The different combinations of dependent possibilities are given by a transition mau-ix. In our weather prediction example, a clear day has a 70°o chance of being followed by another clear day, a ISo chance of a cloudy day, and only a So chance of rain. In Table 2-3, each possibility today is shown on the left, and its probability of changing tomorrow is indicated across the top. Each row totals 100° o, accounting for all weather combinations. The relationship between these events can be shown as a continuous network (see Figure 2-6).

The Markov process can reduce intricate relationships to a simpler form. First, consider a two-state process. Using the markets as an example, what is the probability of an up or down day following an up day, or following a down daj? If there is a 70°o chance of a higher day following a hier day and a 55" <, chance of a higher day following a lower day, what is the probability of any day within an uptrend being up?

Start with either an up or down day, and then calculate the probability of the next day being up or down. This is done easily by simply counting the number of cases, given in Table 2-4a, then dividing to get the percentages, as shown in Table 2-4b.

Because the first day may be designated arbitrarily as up or down, it is an exception to the general rule and, therefore, is given the weight of 50° o. The probability of the second day being up or down is the sum of the joini probabilities

and ihc- Toui-th ilay,

JdJP}. = ( <5-*375 X .VO> I < 3625 X

wliit-l-i can iM>w fcMf se-en lo fcMf converrginff. To iirncrrlzc ttit pirolialiiliiy *jf an up day. look

We can find the chance of an up or down day if the 5-day trend is up simply by substitutine the direction of the 5-day trend (or n-day trend) for the previous days direction in the exanple just given.

Predicting the weather is a more involved case of multiple situations converging and may be very representative of the way prices react to past prices. By approaching the problem in the same manner as the two-state process, a 1 /3 probability is assigned to each situation for the first day; the second days probability is

X 21643 X tO)

P(ralny), = (3663 y. -05) + (-4t6iS X 20) + ( 21645 X AO) = l«Bi5

he general form far snlvirg these Ihree equations is

P{clear)..,= I/«*ar>,K 7<q + IZ-tcloudy), 0 + », x 20J /•(ioudy),. ,= /{tlcar>, >- 5) + [/•{cloudyl, * ) + [/{amv), -40 /{rainy),., = /(cleaO,x 05 + [P{ˆloudy),x 20 + [P{ralny,x .40)

where each i + 1 element can be set equal to the corresponding ith values; there are ihen three equations in three unknowns, which can be solved directly or by matrix multiplication, as shown in Appendix 3 ("Solution to Weather Probabilities Rxpressed as a Markov Chain"). Otherwise, it will be necessarj to use the additional relationship