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Notes for Session 8, Part B


Note 3

Many of the fundamental ideas of probability can be developed from games of chance. In fact, Blaise Pascal, the father of mathematical probability, was inspired in his work by a commission to analyze gambling games.

Mathematical probabilities are used to describe expected frequencies of outcomes that result from repeated trials of "random" experiments. Keep in mind that probabilities are used to describe outcomes of "random" experiments, and that "repeated trials" are an important part of conducting "random" experiments.

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Note 4

If you are working in a group, divide into pairs, play a few rounds of the game, and record the winner. Then pool the groups' results and determine the proportion of wins for Players A or B. This will provide experimental data for judging who might win more often.

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Note 5

In the analysis of this problem, it is crucial to have a clear enumeration of all possible outcomes. There are lots of ways to enumerate the outcomes, and some are more useful than others.

The formal mathematical presentation of the sample space for tossing a pair of dice uses the "order pair" notation. For instance, (2,5) denotes an outcome where one of the dice shows a 2 and the other a 5. This immediately provides a source of confusion for many people. Is (2,3) really different from (3,2)?

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Note 6

From the table of possible outcomes, we can readily see that there are 6 x 6 = 36 pairings of outcomes on the dice.

Once we have completed an enumeration of all possible outcomes, we can assign probabilities to sums or simple outcomes. Remember that the fairness of the dice is crucial to the assignment of probabilities.

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