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Solution to Activity #1:
I hope you changed your mind about playing the game of chance!  Chevalier de Méré did not state the order for the 8 and the 6.  So, there are ten possible ways for him to get a favorable result.  There are only six ways for you to get a favorable result.  I hope you imagined yourself as a wealthy nobleman, because you have a 54.6 percent chance of losing if you accepted to play the game with Chevalier de Méré.

Solution to Activity #5:

Solution to Activity #6:  

(a) Each experiment measures the proportion of mayors in these cities who are Democrats and the proportion who are Republicans during a given term of office.  These proportions depend only on the proportions during the preceding mayoral terms, so the sequence of experiments can be represented by a Markov chain

(b)   P =  

(c)    P² =               P³ =

Solution to Activity #7:
First, look at figure 1.  We need to look at row five of Pascal’s triangle because it represents a combination of five items.  Adding up the sum of the numbers on this row, one should arrive at 32.  This is the total possible number of combinations that can be made with five items.  Next, look at figure 2.  The "g" represents the number of girls and the "b" represents the number of boys.  The binomial expansion of Pascal’ triangle shows the possible ways of getting each combination of items.  For example, "5g⁴b" means that in a family of five there are five ways (5) of getting four girls (g⁴) and one boy (b).  

Figure 1

Figure 2

(b) Looking at row five of Pascal’s triangle, one sees that there are 32 possible ways to choose five boys and girls, but there are five ways to choose four girls and one boy.  Therefore, the probability is 5/32.

(c)  Looking at row five of Pascal’s triangle, one sees that there are 32 possible ways to choose five boys and girls, but there are ten ways to choose three girls and two boys.  Therefore, the probability is 10/32, which reduces to 5/16.

(d)  Looking at row five of Pascal’s triangle, one sees that there are 32 possible ways to choose five boys and girls, but there are ten ways to choose two girls and three boys.  Therefore, the probability is 10/32, which reduces to 5/16.

(e)  Looking at row five of Pascal’s triangle, one sees that there are 32 possible ways to choose five boys and girls, but there are five ways to choose one girl and four boys.  Therefore, the probability is 5/32. Looking at row five of Pascal’s triangle, one sees that there are 32 possible ways to choose five boys and girls, but there is only one way to choose five boys.  Therefore, the probability is 1/32.