# Stochastic Processes in Physics and Chemistry

Chapter01 - Home
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Exercise. This is the solution to exercise 1.1.1 in the book.

Solution. If the faces of the die are unbiased then each of the six faces has an equal probability of occurrence. Thus the range of the number of points is:

${\Omega }_{X}=\left\{1,2,\dots ,6\right\}$

and the probability distribution is:

$p\left(X=1\right)=p\left(X=2\right)=\dots =p\left(X=6\right)=\frac{1}{6}$

For two dice the range of values are pairs $\left\{m,n\right\}$. The range of values is given by:

${\Omega }_{{X}^{2}}=\left\{\left[1,1\right],\dots ,\left[1,6\right],\dots ,\left[6,1\right],\dots ,\left[6,6\right]\right\}$

Thus there are a total of 36 pairs. If the dice are unbiased, then each outcome is equally probable. Thus the probability of each element in the set ${\Omega }_{{X}^{2}}$ occurring is:

$p\left(X=\left[i,j\right]\right)=\frac{1}{36},\phantom{\rule{1em}{0ex}}i,j\in \left\{1,\dots ,6\right\}$