ODDS & ENDS... (this page will be updated periodically and may be at its best when viewed with Internet Explorer)

Odds are...

Odds and probabilities are closely related. If, for example, a horse named Man is given a probability of 30% to win a particular race, then presumably he would lose with probability 70%. We say that the odds of Man's winning are 3:7 -- that is 30 to 70, reduced to a pair of relatively prime integers.

In general, the odds in favor of an outcome are given as a:b, simplified to the smallest integers. If so, then the probability of that outcome is stated as a/(a+b) and the probability against that outcome is b/(a+b).

Practice exercise: The odds of the Jets winning the playoffs are 14:11. What is (a) P(Jets win) =  ?  (b) P(Jets lose) =  ?

Quite Probably...

Notation: We abbreviate the phrases , "the probability of getting a 3 on a roll of a die," "the probability of getting a 3 or better," and "the probability of rain," as P(X=3), P(X>3), and P(Rain), respectively. The parenthesized item is an event and each of these statements has a value, e.g., P(X>3)=2/3, that must obey the probability rules that we discussed. If E is some chance event, then P(E) is the relative frequency with which E would occur in a large number of experiments.

...and Relatively Frequently...

Notation: Suppose that we roll a die n=10 times and obtain: {6 2 4 1 4 2 2 6 2 4}. We abbreviate the phrase , "we got a '6' twice, a '4' three times, and a '2' four times"  as Freq(X=6)=2, Freq(X=4)=3, and Freq(X=2)=4, respectively. These "frequencies" can be abbreviated as F(6)=2, F(4)=3, etc. The sum of all frequencies, FSUM = F(1)+...+F(6) = 10, equals n, the number of numbers in the original set. We may obtain the "relative frequencies" by dividing every F by FSUM. Thus, f(1)=0.1, f(2)=.4, etc. The sum of all relative frequencies, fSUM = f(1)+...+f(6) = 1 (always equals 1). Relative frequencies are used to estimate probabilities (e.g., if we didn't know the probability of rolling a '6', we might estimate it as P(X=6)≈f(6)=.2. The larger n, of course, the better the approximation...
On relative & cumulative frequency distributions

The Monty Hall Game: Let's Make a Deal...

A simple probability problem has provoked a series of disputes between a columnist said to be the holder of the world's highest IQ score and numerous Ph.D.s in academia since it appeared in the "Ask Marilyn" column in the September 9, 1990 issue of Parade Magazine. The original three door game show problem appeared in the column is stated as follows: "Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice? - D. Craig F. Whitaker, Columbia, Md." (http://isds.bus.lsu.edu/chun/teach/reading-a/mypaper.htm)

Games of Chance & "Expectations" of Profit

 Is there a way to compare two games or lotteries against one another? Suppose that lottery A costs $1/ticket and has 3 prizes of $1000, $5,000 and $100,000 (with respective probabilities of 1 in 5000, 1 in 25,000 and 1 in 500,000). Lottery B also costs $1 but has only one jackpot of $10,000,that comes up with one chance in 12,500. Which is a better deal and if so, in what sense? [Hint: A game between 2 players is said to be fair if E(X)=0 where X is the amount player Smith wins from his opponent in each play. If X is negative it is a loss to Smith. If E(X)< 0, the game is unfair to Smith. If Smith must pay Jones $T at each play (as in a lottery), then the criterion becomes E(X)-$T. If this is < 0, the game is unfair to the player which is almost always what happens in casinos and lotteries, but you may still want to compare 2 games/lotteries to determine which is less unfair.]

About Simulation

 You may have tried your hand at flying a plane on your computer screen using Flight Simulator or some such software. As the term suggests, a simulation mimics the behavior of some system (such as a plane) without using the system at all. What you use is a facsimile of the system, i.e., a model. The more sophisticated the model, the more realistic the simulation usually is. Various methods and software can be used. We use MS Excel, because it is widely available and relatively easy to use as a simulation laboratory. You must use some special functions, like those listed on the BRM website, in order to introduce "chance" behavior for your model.

... definitions: 

System: the mechanism to be studied (the flip of a coin, a waiting line at a bank, the weather, the Economy)

Simulation model: a description of the essential relationships of the system (tables, equations, “if-then” conditions, etc.)

Simulation:  A very effective, Make-Believe process that uses random numbers to imitate any chance behavior in the model.

Objective: Usually, to design/re-engineer/study the system quickly, inexpensively &, perhaps,  before it is even built.

 Related terms: Digital simulation, Excel simulation, Graphical simulation, Real-time simulation (video games, virtual reality, training?)

Calculating population mean, variance, & standard deviation

The Bell Curve and Excel:

The following Excel functions enable us to compute probabilities that pertain to normal distributions and to simulate related models on the computer. The N...DIST are "forward" functions. You give them any arbitrary value of the random variable and they return a probability (except for the "FALSE" case, below). The N...INV are "backward" functions. You give them a probability and they return a percentile (numerical value for the random variable). The format of these functions is the following.

NORMSDIST(Z): Provides the value of the cumulative standard normal, N(0,1), for the specified Z-value
.......Ex: The first point of the 3-point Rule, is
.......NORMSDIST (1) - NORMSDIST (-1) è 0.84134474 - 0.15865526 = 0.68268948 or about 68%

NORMDIST(X, Mean, StandardDeviation, Cumulative): Provides the value of the normal, N(mean, std dev), for the given value X. If you want the cumulative make the 4th parameter "TRUE"; for the regular normal, make it "FALSE";
.......Ex: If X~N(7,2) then NORMDIST(7, 7, 2, TRUE) = 0.5 (area under the "bell" to the left of X=7)
.......whereas, NORMDIST(7, 7, 2, FALSE) = 0.2 (the height of the "bell" at X=7)

NORMSINV(Y): Gives the inverse of the std normal N(0,1), for a given value Y
.......Ex: NORMSINV (0.84134474) è 1

NORMSINV(RAND()): We may use this to generate random numbers according std normal N(0,1)

NORMSINV() gives Z values.  To convert to X values, say, for  X~ N(7,2), we would solve the defining equation  Z=(X-7)/2  for X; that is,  X=2Z+7. Therefore, to generate random numbers that have an arbitrary normal distribution, for the purposes of a simulation, we can use the Excel formula:.......=2* NORMSINV(RAND()) + 7 ...
...and, for the general N(Mean,SD), we can use... =SD* NORMSINV(RAND()) + Mean

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