Rolling dice, probability

We should note that this relationship applies for the case at hand, but it is not general. The factor of 1 in the numerator is present because we are looking for a single specific roll (of a number four) on each die and the six in the denominator is there because there are six possible rolls for each die. With this relationship, however, we could easily predict that the chances of rolling the same number with five dice in one roll are one in 1296. (Note that the chances of rolling a specified number on all five dice—say all fours—are 1 in 7776. But if we do not specify in advance which of the six possible numbers we want on all five dice, then there will be six possible outcomes instead of just one.) Our experience with rolling dice is that we expect to have some random assortment of numbers present when five dice are rolled. Why There are very many ways to obtain a random roll. Such a roll occurs far more often precisely because it is more probable. 

Preassigned Probability When the likelihood of all possible outcomes of a given event is known or can be determined, the probability of such outcomes are said to be preassigned (rolling dice, tossing a coin, etc.). 

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

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... 

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

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) ... 

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 ... 

In the roll of two dice, what total number is the most likely to occur Is there an energy reason why this number is favored Would energy have to be spent to increase the probability of getting a particular number (that is, to cheat) ... 

The uncertainty principle has profound implications for an atomic model. It means that we cannot assign fixed paths for electrons, such as the circular orbits of Bohr s model. As you ll see next, the most we can ever hope to know is the probability—the odds—of finding an electron in a given region of space but we are not sure it is there any more than a gambler is sure of the next roll of the dice. 

An analogy here is useful the probability of rolling a given number with a six-sided die is 1 /6. The probability that the same number will appear on n dice rolled simultaneously is (1/6) . 

Probability does not necessarily change just because an event occurs. The likelihood of rolling a six on a dice is one in six. Just because we happen to roll a six on our next turn does not change that probabiUty. 

Problem A pair of fair (well balanced) dice is rolled. What is the probability of getting snake eyes -- one dot on each die ... 

Figure 10.3 I The probability of rolling a given total value on a pair of dice depends on the number of different combinations that produce that total. The least likely rolls are 2 and 12, for example, because there is only one possible combination that gives each of those totals. The most likely total is seven because there are six different rolls that add up to that number. (Note that for rolls in which the two individual dice show different values, two possibilities exist. For a total roll of three, for example, the two combinations would be 1, 2 and 2, 1.)...

Some games include dice with more than six sides. If you roll two eight-sided dice, with faces numbered one through eight, what is the probabUity of rolhng two eights What is the most probable roU ... 

If we increase the number of dice, the distribution in probability among the macrostates becomes much more dramatic. For example, with six dice there are 46,656 (or 6 ) possible miaostates ranging from 6 to 36, for a total of 31 macrostates. For the macrostates 6 and 36 there is only one possible microstate [(1,1,1,1,1,1) and (6,6,6,6,6,6), respectively], so they both have W = and a probability of 1/46,656. There are, however, 4332 possible ways of rolling a 21 (the most probable macrostate) giving a probability of 4332/46,656. Thus, we are over 4000 times more likely to roll a 21 than a 6 or 36. As we increase the number of dice, the probability of the most probable macrostate increases rapidly relative to less probable macrostates... 

When rolling four dice, consider the microstates as corresponding to the individual numbers on each die and the macrostates as corresponding to the sum of these numbers, (a) How many microstates and macrostates are they (b) What are the most probable and least probable macrostates, (c) What is the ratio of the number of microstates corresponding to the most probable macrostate versus the least probable macrostate. 

The law of addition says that if an event can happen in a number of different ways, then the probability of the event happening is obtained by adding up the probabilities of each of the separate ways. Thus in calculating the probability of getting an even number with one roll of a dice, it is known that the probability of each number is Ve (since there are six possible numbers), and the event wanted (an even number) can occur in three possible ways (for numbers 2, 4 or 6). So by the law of addition, the probability is obtained by adding the separate probabilities. In this case it will be Ve + Ve + Ve = Vi) therefore probability P = 0.5. 

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