Counting the number of outcomes in games of chancehas been a popular pastime for many centuries. This wasof interest not only because of the gambling that wasinvolved, but also because the outcomes of games ofchance were often interpreted as divine intent. Thus, itwas just about a thousand years ago that a bishop in whatis now Belgium determined that there are 56 different ways in which three dice can fall provided one is inter-ested only in the overall result and not in which die does what. He assigned a virtue to each of these possibilitiesand each sinner had to concentrate for some time on thevirtue that corresponded to his cast of the dice.(a) Find the number of ways in which three dice can allcome up with the same number of points.(b) Find the number of ways in which two of the threedice can come up with the same number of points, whilethe third comes up with a different number of points.(c) Find the number of ways in which all three of the dicecan come up with a different number of points.(d) Use the results of parts (a), (b), and (c) to verifythe bishop’s calculations that there are altogether 56possibilities.

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Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.