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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

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

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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