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					                                 Teacher Version
                                                                                 TI-89 Version
                                      Probability
                                      The Game

Kennedy and Aidan are playing a game. They take turns rolling a fair six-sided die. The
first one to roll six wins the game.

1. If Kennedy rolls first, what is the probability that she wins the game in one roll?
    1
    6

2. If Kennedy rolls first, what is the probability that she wins the game on her second roll?
   In order for Kennedy to get a second roll, she would have to roll something other than a
   six (lose) on her first roll and Aidan would also have to lose on his first roll, then
   Kennedy would roll a six.
    5 5 1 25
        
    6 6 6 216

3. If Kennedy rolls first, what is the probability that she wins the game on her third roll?
   Kennedy loses, Aidan loses, Kennedy loses, Aidan loses, Kennedy Wins
    5 5 5 5 1 625
                        0.08038
    6 6 6 6 6 7776

4. Write a recursive definition to find the probability that Kennedy wins on her nth roll?

         1
   p1=
         6

      5 5          25
   pn=   pn 1     pn 1
      6 6          36




                                          Page 1

                                                                                Rita Grunloh
                                                                  rgrunloh@lexington.k12.il.us
                                 Teacher Version
                                                                                  TI-89 Version

5. Set up a Table in your calculator using the recursive definition above.
       a. Your calculator should be in sequence graphing mode (Mode – Graph –
           Sequence)
       b. In the [Y=] (F1) enter your recursive definition in u1=
       c. Enter your initial value in ui1=
       d. Make a table and complete the chart below.
               i. Use [TBLSET] (F4)and start your table at 1 ( Tblstart =1) with a change
                  of 1 ( tbl =1)
              ii. Use the [TABLE] (F5) function to display your table

                   Number of          Probability of
                      Rolls          Winning on the
                       (n)               nth Roll
                        1                 .16667
                        2                 .11574
                        3                 .08038
                        4                 .05582
                        5                 .03876
                        6                 .02692
                       10                 .00626
                       20                 .00016
              iii. Use this table to check your answers to #1-3 on this worksheet.

6. Look at the probability function again. Is there a pattern (it will help to write out each
   probability and look for a pattern before you simplify). Write an equation (non-
   recursive) for the probability that Kennedy will win on her nth roll.
                         n1
              1  25                                                     1         25
           p1=                This is a geometric sequence, with a1      and r 
              6  36                                                     6         36

7. Is there a name for this type of sequence? What is it?
                                            1          25
   This is a geometric sequence, with a1     and r 
                                            6          36

8. If Kennedy rolls first, what is the probability that she will win the game no later than her
   third roll?
   The probability that Kennedy wins no later than her third roll is equal to sum of the
   probabilities that she wins on her first roll + the probability that she wins on her second
   roll + the probability that she wins on her third roll
    1 25 625 2849
                        
    6 216 776 2619
   .16667  .11574  .08038  .36278

                                           Page 2

                                                                                Rita Grunloh
                                                                  rgrunloh@lexington.k12.il.us
                                Teacher Version
                                                                               TI-89 Version

9. Write a recursive definition for the probability that Kennedy will win the game no later
   than her nth roll.

                        n 1
             1  25 
   Sn= Sn1   
             6  36 

10. Make a table in your calculator and complete the chart below.


                   Number of        Probability of
                     Rolls         Winning no later
                      (n)             than the
                                      nth Roll
                           1           .16667
                           2           .28241
                           3           .36278
                           4            .4186
                           5           .45736
                          10           .53123
                          20           .54508
                          25           .54539
                          30           .54544


11. What did you notice? Make a prediction on the probability that Kennedy will win the
    game (no matter how long it takes)? Explain your answer.

   Students should notice that the sum of the probabilities is getting closer to .54545 .
   This should lead the students to the conclusion that the probability that Kennedy wins
   will be .54545. They may also notice that the probability that someone wins on the nth
   roll gets smaller and smaller as n gets larger. Through discussion, it should be noticed
   that this is an infinite geometric sequence and the probability that Kennedy wins is the
                                                                         a
   sum of the infinite geometric sequence given by the formula S  1 .
                                                                        1 r




                                         Page 3

                                                                              Rita Grunloh
                                                                rgrunloh@lexington.k12.il.us
                                Teacher Version
                                                                                TI-89 Version
                                     Probability
                                     The Game
                                     Homework

1. If Kennedy and Aidan play the same game, what is the probability that Kennedy wins if
   Aidan goes first?

       5 1 5
    a1  
       6 6 36
      5 5 25
    r  
      6 6 36

                  5
          a1           5
    S        36   .4545
         1  r 1  25 11
                   36


2. Suppose that Amy, Baron, and Cassy are taking turns spinning a spinner. Each player
   has chosen a color on the spinner. Each player will win only if their chosen color comes
                                                                                    1 3
   up on during their turn. The probability that each player’s color will come up is ,    ,
                                                                                    5 10
        3
   and     for Amy, Baron, and Cassy, respectively. Amy spins first, then Baron, then
        7
   Cassy, and the game continues until one player wins. What is the probability that Cassy
   wins?

   The first term is the probability that Cassy wins on her first roll. This will happen if Amy
                                                               4 7 3 6
   does not win, Baron does not win, and Cassy wins: a1    
                                                               5 10 7 25
   The common ratio is the probability that they each lose, thus moving them all into the
                       4 7 4 8
   next round: r    
                       5 10 7 25

                                                    6
                                            a           6
   The probability that Cassy wins is: S  1  25        .3529
                                           1  r 1  8 17
                                                     25



                                         Page 4

                                                                              Rita Grunloh
                                                                rgrunloh@lexington.k12.il.us
                                Teacher Version
                                                                             TI-89 Version
3. Take another look at problem number 2, what is the probability that Cassy wins if she
   goes first? What if she goes second? Does it matter what order the other two players
   go? Explain your answers.

                                                                3
   If Cassy goes first the first term of the sequence would be    . The common ratio would
                                                                7
   remain the same, the probability that the other players each lose between Cassy’s turns
                                              4 7 4 8
   and then Cassy winson her next turn, r                 . The probability that Cassy
                                              5 10 7 25
   wins if she goes first is:
                  3
          a1               75
    S        7             .6303 In this case, it does not matter what order the other
         1  r 1  8 119
                   25
   two players go.

                                                                            4 3 12
   If Cassy goes second (after Amy) the first term of the sequence would be         .
                                                                            5 7 35
   The common ratio would remain the same, the probability that the other players each lose
                                                                  4 7 4 8
   between Cassy’s turns and then Cassy winson her next turn, r               . The
                                                                  5 10 7 25
   probability that Cassy wins if she goes second (after Amy) is:
                  12
         a1              60
   S         35           0.5042
        1  r 1  8 119
                    25

                                                                3 3 9
   If Cassy goes second (after Baron) the first term will be         . The probability
                                                               10 7 70
   that Cassy wins if she goes second (after Baron) is:
                  9
          a1              45
    S        70           0.1891
         1  r 1  8 238
                   25

   The only time the order matters is in determining the first term of the sequence.




                                         Page 5

                                                                               Rita Grunloh
                                                                 rgrunloh@lexington.k12.il.us

				
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