SOLUTION: Consider a game in which you roll a pair of dice. Suppose you are paid $7.20 if you roll a pair and nothing otherwise. If you are asked to pay $1 to play this game, is it a fair

Question 536768: Consider a game in which you roll a pair of dice. Suppose you are paid $7.20 if you roll a pair and nothing otherwise.
If you are asked to pay $1 to play this game, is it a fair game? If it is not fair, is it unfair to your benefit or detriment? Explain your choice. Develop a tree diagram for tossing two, eight-sided gaming dice to figure out how many possibilities there are. Discuss the purpose of using such a visual in working out probability.
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!
Consider a game in which you roll a pair of dice. Suppose you are paid $7.20 if you roll a pair and nothing otherwise.
If you are asked to pay $1 to play this game, is it a fair game? If it is not fair, is it unfair to your benefit or detriment? Explain your choice.

Event         Winnings = X      Prob.   Expectation =             

Pair      $7.20-$1.00 = $6.20    1/6    $6.25�(1/6) = $1.03333         
Non-pair        -$1.00           5/6   -$1.00�(5/6) =  -.83333
--------------------------------------------------------------                                                      $ .20

It is unfair to your benefit, as you will average winning 20 cents per game. 


Develop a tree diagram for tossing two, eight-sided gaming dice to figure out
how many possibilities there are. That's a big tree diagram!  Here it is partly
drawn. You'll have to draw in all the many missing lines connecting all the
numbers. There are 64 possibilities.

                                                      /1
                                                     / 2
                                                    /  3
                                                  /    4
                                                1      5
                                               /  \    6
                                             /      \  7
                                            /         \8
                                          /    
                                         /            /1
                                       /            /  2
                                      /            /   3
                                    /             /    4
                                  /              2     5
                                 /                \    6
                               /                    \  7
                             /                        \8
                            /                     
                          /                           /1
                        /                           /  2
                       /                           /   3
                     /                            /    4
                   /                             3     5
                 /                                \    6
               /                                    \  7
              /                                       \8
            /
          /                                           /1
        /                                           /  2
      /                                            /   3
     /                                            /    4
   /                                            4      5
  \                                               \    6
   \                                                \  7
     \                                                \8

                                                      /1
                                                    /  2
                                                   /   3
                                                  /    4
                                                5      5
                                                  \    6
                                                    \  7
                                                      \8

                                                      /1
                                                    /  2
                                                   /   3
                                                  /    4
                                                6      5
                                                  \    6
                                                    \  7
                                                      \8

                                                     / 1
                                                    /  2
                                                   /   3
                                                  /    4
                                                7      5
                                                  \    6
                                                    \  7
                                                      \8

                                                     / 1
                                                    /  2
                                                   /   3
                                                  /    4
                                                8      5
                                                  \    6
                                                    \  7
                                                      \8



The advantage of a tree diagram is that you can see the branching visually
and why you multiply.  They show the complete sample space and it is easy to
calculate the probabilities without missing any outcomes. The disadvantage is
that they are hard to draw, and when there are many branches, the tree diagram
would be very huge.  

Edwin

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