Dice throws

Cratering Data Report , AFWL-TR-76-19, Air Force Weapons Lab, Kirtland AFB, NM (1976) (Limited distrib) 152) J. Wisotski, Middle North Series. Pre-Dice Throw I and II Events, Technical Photography , Final Rept, Denver Res Inst, Denver (1976) (Limited distrib)... 

North Series. Pre-Dice Throw 1 and II Events, Technical Photography , Final Rept, Denver Res Inst, Denver (1976) (Limited distrib)... 

State of system always same after each experiment i.e state highly repeatable (deflection of stable beam) 1 decreasing repeatability of state of system 1 State of system often different after each experiment i.e state not highly repeatable (dice throw) 1 1... 

Example 15.3. Find the variance for the distribution of dice throws in the previous example... 

The standard deviation for our dice throw example is equal to... 

For most distributions with a single peak, about two-thirds of the members lie within one standard deviation of the mean. In this case, six of the 36 possible dice throws lie below 4.6 and six of the possible throws lie above 9.4, so that 24 of the throws (exactly two-thirds) lie within the range from (n) - a and (n) + an. 

Example 5 Probability Calculation If a six-sided die marked with the mimhers 1, 2, 3, 4, 5, and 6 is thrown, the prohahihty that any given mimher will he uppermost is F6. If the die is thrown twice in succession, then the proh-ahility of a given sequence of mimhers occurring, say, 5 followed hy 6, is (F6)(l/6) = 1/36. The chance of any particular mimher occurring 0, 1, 2, 3, or 4 times in four throws of the die (or in a simultaneous throw of four dice) is given hy the successive terms of Eq. (9-77), expanded as... 

As Tribus, 1969, says, all probabilities are conditional. In the example of the dree, the probabilities are conditioned on the assumption that the dice are perfect and the method of throwing has no effect on the outcome. Some writers (e.g., deMorgan, 1847) say, probability refers to the belief by a mind having uncertain knowledge. This is the interpretation of probability in the Zion-Indian Point (ZIP) and some other PSAs. IVobabiiity in this sense attempts to include all information e.g., QA that could affect the performance of a piece of equipment. Such information may be conveyed as a distribution whose height is proportional to confidence in the belief and who.se width reflects uncertainty (refer to Section 2.6). 

To illustrate what these statements mean, consider a pastime far removed from chemistry tossing dice. If you ve ever shot craps (and maybe even if you haven t), you know that when a pair of dice is thrown, a 7 is much more likely to come up than a 12. Figure 17.1 shows why this is the case. There are six different ways to throw a 7 and only one way to throw a 12. Over time, dice will come up 7 six times as often as 12. A 7 is a state of high probability a 12 is a state of low probability. 

A random demand is not sporadic with respect to a period length if we can expect that the outcome for a period is almost surely greater than zero. In other words 8 (0) = 0 which does not mean that the outcome 0 is impossible if you throw a dice with infinitely many faces then any outcome has probability 0. 

The National Science Foundation (NSF) Program in Geography and Regional Science funded much of the research described in this volume (Project 0095993). I am deeply indebted to the NSF and several anonymous reviewers for being willing to throw the dice on this project. 

This classical problem may help to illuminate this formalism Consider a die. When it is thrown there are six possible results i where 1 [Pg.338]

Ad a. To establish the a priori distribution one has to take into account the actual system. It turns out that many systems have a level of description where a simple guess for the probability distribution can be made. In most cases this amounts to identifying units with equal probability. When throwing two dice one computes the a posteriori distribution of the total number of points from the assumed a priori distribution made up by equal probabilities for the 36 elementary events. There are good reasons for this assumption, but as always in physics it has to be verified by experiment no amount of mathematics can show that a die is not loaded. 

The loaded dice will not show the winning number at every throw but will do so rather more frequently than an unmodified one. That is precisely what happens in chemistry where complex reactions always yield an approximately normal distribution around... 

Favourable number of ways to event E Total number of ways of the experiment Problem 43 Find the chance of throwing more than 4 in one throw of cubic dice marked 1 to 6 its six faces. 

Solution Here are 6 equally likely cases of which only 2 are favourable because we want 5 or 6 on the upper face of the cubical dice. Hence, the required probability of throwing more than 4 in one throw with one dice... 

Problem 44. In a single throw with two dices, what is the probability of throwing 9 ... 

The dynamic nature of molecules can be troublesome for students. To aid in understanding molecular behavior, a number of games mimic the activities that chemical compounds may undergo. To duplicate chemical equilibrium, students throw dice in one published game (Edmonson and Lewis 1999). By participating in the equilibrium firsthand, students may better comprehend the system. 

After millennia of debating whether nature is in its essence deterministic, and therefore predictable, the answer is still not known. Even Albert Einstein who pioneered work in statistical mechanics was quoted (1) as saying "Quantum mechanics is very impressive. But an inner voice tells me that this is not the real Jacob. The theory has much to offer, but it does not bring us closer to the secret of the Old One. At least I am convinced that He does not throw dice."... 

Itis important to note that the uncertainty in measurement Tm talking about is not due to scientists just not having sensitive enough equipment. No matter how sensitive the equipment, it is a fundamental fact of nature that we cannot get exact information about what small-scale systems are doing and where they are. In other words, nature will always keep us somewhat in the dark about these measurements. This limitation has bothered many scientists for many years. Even Albert Einstein didn t accept it at first. In referring to the probabilistic nature of the atomic world, in a letter to fellow physicist Max Born, Einstein stated, "I, at any rate, am convinced that He [meaning God] does not throw dice." ... 

And a quick note if you use dice instead of pennies or M M S Each time you throw the dice on the surface remove just the dice that are showing two or the dice showing six (or any other number you choose). By removing on average, only one-sixth of the dice, the procedure will take longer. 

Example 3. For the throw of two fair dice (i.e. each number has 1/6 probability of occurring), let jr be a random variable that is equal to the sum of the numbers then... 

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