Dice, probability

As an example, the possible outcomes and associated probabilities for rolling a pair of six-sided dice are... 

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). 

Certain "states" are more probable than others. For example, when you toss a pair of dice, a 7 is much more likely to come up than a 12. 

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. 

Almost everybody is familiar with the notion of probability as applied to games of chance. When discussing dice games, we assign the probability 1/6 to the event a one comes up because there are six faces of the die that are equally likely to come up and the desired face is, therefore, only one possibility out of six. Operationally, the number 1/6 is taken to mean that, in a long sequence of N tosses of the die, approximately 1/6 -N ones will show up. In other words, the... 

Reluming now to a game, consider the possible sum of numbers on two dice. Each die has six sides and, assuming that the dice are not loaded, the outcome of the roll of each die has six equally probable possibilities. The probability for each number on a given die is thus equal to 1/6 and the combined probability for the dice is equal to 1/36. The possible values of the sum and the corresponding probabilities are then given in Table 5. These results are plotted in Fig. 1, which represents the probability distribution for this example. 

The net result for the probability distribution of the sum of the numbers on two dice is then represented in Fig. 1. The well-known significance of the number seven becomes evident, as it has the greatest probability. As a second roll of the dice is independent of the result of the first, the chances of getting an eleven is only 2/36. However, it is the conditional probability, that is, the... 

Werner Heisenberg stated that the exact location of an electron could not be determined. All measuring technigues would necessarily remove the electron from its normal environment. This uncertainty principle meant that only a population probability could be determined. Otherwise coincidence was the determining factor. Einstein did not want to accept this consequence ("God does not play dice"). Finally, Erwin Schrodinger formulated the electron wave function to describe this population space or probability density. This equation, particularly through the work of Max Born, led to the so-called "orbitals". These have a completely different appearance to the clear orbits of Bohr. 

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. 

When rolling two dice, what is the probability of rolling a sum of eight ... 

You can assume, in probability problems, that all outcomes occur at random, unless otherwise noted. If the events described concern dice, assume that the dice always lands flat on a number. If the events concern a spinner, assume that the spinner never lands on a dividing line. 

Probability problems concerning two dice rolled are common examples on tests. It is helpful to make a table of all possible outcomes as shown below ... 

By trying to address all of these issues in parallel, our aim is to increase the probability that any particular screening campaign will be successful. The factors that influence the outcome of each roll of the HTS dice are multifactorial, and altering the odds in our favor for each of these factors is the purpose of much of the methods-development activities currently carried out in most research-based pharmaceutical companies. Only by understanding the odds of different strategies and components can one build an approach that balances long and short odds. 

The Problem What is the probability that, when you roll two dice, you ll get a sum of 6 ... 

The Problem When you roll two dice, what is the probability that the sum will be even or that the roll will be a doubles (both faces the same) ... 

Exercise. Let X be the number of points obtained by casting a die. Give its range and probability distribution. Same question for casting two dice. 

Exercise. Two dice are thrown and the outcome is 9. What is the probability distribution of the points on the first die conditional on this given total Why is this result not incompatible with the obvious fact that the two dice are independent ... 

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 idea of equal probabilities has been elevated by Laplace510 to the rank of a philosophical principle, called principle of insufficient reason . Like many philosophical principles it leaves the essential question unanswered How do I select the elementary events to which equal a priori probabilities are to be assigned In textbook problems about tossing dice or drawing cards it is obvious what the author has in mind. One knows that he is concerned with the mathematics of step b and that the dice and cards merely serve as a ritual way of defining an a priori distribution. In actual applications, however, step a cannot be dismissed so cavalierly. 

When rolling a pair of dice, there are two ways to get a point total of 3 (1 + 2 2 + 1) but only one way to get a point total of 2 (1 + 1). How many ways are there of getting point totals of 4-12 What is the most probable point total ... 

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 49. A pair of dice is thrown. What is the probability of getting a total of 7 ... 

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