Difference between revisions of "1993 AIME Problems/Problem 11"
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== Solution == | == Solution == | ||
| − | {{ | + | The probability that the <math>n</math>th flip in each game occurs and is a head is <math>\frac{1}{2^n}</math>. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the first person winning the game is <math>\frac{1}{2}+\frac{1}{8}+\frac{1}{32}+\cdots = \frac{\frac{1}{2}}{1-\frac{1}{4}}=\frac{2}{3}</math>, and the probability of the second person winning is <math>\frac{1}{3}</math>. |
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| + | Let <math>a_n</math> be the probability that Alice wins the <math>n</math>th game, and let <math>b_n</math> be the probability that Bonnie wins the <math>n</math>th game. | ||
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| + | If Alfred wins the <math>n</math>th game, then the probability that Alfred wins the <math>n+1</math>th game is <math>\frac{1}{3}</math>. If Bonnie wins the <math>n</math>th game, then the probability that Alfred wins the <math>n+1</math>th game is <math>\frac{2}{3}</math>. | ||
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| + | Thus, <math>a_{n+1}=\frac{1}{3}a_n+\frac{2}{3}b_n</math>. | ||
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| + | Similarly, <math>b_{n+1}=\frac{2}{3}a_n+\frac{1}{3}b_n</math>. | ||
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| + | Since Alfred goes first in the <math>1</math>st game, <math>(a_1,b_1)=\left(\frac{2}{3}, \frac{1}{3}\right)</math>. | ||
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| + | Using these [[recursive]] equations: | ||
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| + | <math>(a_2,b_2)=\left(\frac{4}{9}, \frac{5}{9}\right)</math> | ||
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| + | <math>(a_3,b_3)=\left(\frac{14}{27}, \frac{13}{27}\right)</math> | ||
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| + | <math>(a_4,b_4)=\left(\frac{40}{81}, \frac{41}{81}\right)</math> | ||
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| + | <math>(a_5,b_5)=\left(\frac{122}{243}, \frac{121}{243}\right)</math> | ||
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| + | <math>(a_6,b_6)=\left(\frac{364}{729}, \frac{365}{729}\right)</math> | ||
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| + | Since <math>a_6=\frac{364}{729}</math>, <math>m+n = 1093 \equiv \boxed{093} \pmod{1000}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=1993|num-b=10|num-a=12}} | {{AIME box|year=1993|num-b=10|num-a=12}} | ||
Revision as of 22:24, 1 July 2008
Problem
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is 
, where 
and 
are relatively prime positive integers. What are the last three digits of 
?
Solution
The probability that the 
th flip in each game occurs and is a head is 
. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the first person winning the game is 
, and the probability of the second person winning is 
.
Let 
be the probability that Alice wins the th game, and let 
be the probability that Bonnie wins the th game.
If Alfred wins the th game, then the probability that Alfred wins the 
th game is . If Bonnie wins the th game, then the probability that Alfred wins the th game is 
.
Thus, 
.
Similarly, 
.
Since Alfred goes first in the 
st game, 
.
Using these recursive equations:
Since 
, 
.
See also
| 1993 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
