In order to derive the rules of probability, consider the slightly more general example shown in Figure 1.10 involving two random variables X and Y (which could for instance be the Box and Fruit variables considered above). We shall suppose that X can take any of the values xi where i =1,...,M, and Y can take the values yj where j =1,...,L. Consider a total of N trials in which we sample both of the variables X and Y , and let the number of such trials in which X = xi and Y = yj be nij. Also, let the number of trials in which X takes the value xi (irrespective of the value that Y takes) be denoted by ci, and similarly let the number of trials in which Y takes the value yj be denoted by rj.

为了引出概率的规则，假设一个更一般化的例子，如图1.10,有两个随机变量X和Y。我们会假设X可以取任何的xi，其中i=1,...,M，；Y能取任何yj，其中j=1,...,L.假设进行了N次试验，其中我们抽样两个随机变量X和Y，让X=xi并且Y=yj的次数是nij。并且，X是xi的次数，记为ci，类似，Y=yj的次数，记为rj。

The probability that X will take the value xi and Y will take the value yj is written p(X = xi, Y = yj ) and is called the joint probability of X = xi and

X取Y取得概率记作p(X= Y=)，即叫做X=Y=的联合概率。

Y = yj . It is given by the number of points falling in the cell i,j as a fraction of the total number of points, and hence

它就是落在i,j空格里的点的个数和所有点总数的比率。因此有

Here we are implicitly considering the limit N → ∞. Similarly, the probability that X takes the value xi irrespective of the value of Y is written as p(X = ) and is  given by the fraction of the total number of points that fall in column i, so that Because the number of instances in column i in Figure 1.10 is just the sum of the number of instances in each cell of that column, we have ci =   j nij and therefore,

在这里我们考虑N趋于无群大，相似的，不管Y取什么，p(X=)的概率是落入i列，即c i= n ij，因此

from (1.5) and (1.6), we have

结合公式1.5和1.6，我们有

which is the sum rule of probability. Note that p(X = xi) is sometimes called the marginal probability, because it is obtained by marginalizing, or summing out, the other variables (in this case Y ). If we consider only those instances for which X = xi, then the fraction of such instances for which Y = yj is written p(Y = yj |X = xi) and is called the conditional probability of Y = yj given X = xi. It is obtained by finding the fraction of those points in column i that fall in cell i,j and hence is given by

这就是概率的加法规则。注意P(X=xi)有时也叫做边际概率，因为它是通过边缘化或者加和了其他变量得到的，如果我们考虑仅当X=xi时的情况，那在这种情况下Y=yj的部分，记作当X=xi时，Y=yj的条件概率。它是在i列中落在空格ij的部分，公式如下

From (1.5), (1.6), and (1.8), we can then derive the following relationship

结合1.5 1.6和1.8，我们能推到出下面的关系，

which is the product rule of probability.

这是概率的乘法规则。