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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. n1 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= Sn1 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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posted: | 11/12/2011 |

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