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Difference between revisions of "2005 AMC 10A Problems/Problem 18"

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Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?  
 
Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?  
  
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<math> \mathrm{(A) \ } \frac{1}{5}\qquad \mathrm{(B) \ }  \frac{1}{4}\qquad \mathrm{(C) \ }  \frac{1}{3}\qquad \mathrm{(D) \ }  \frac{1}{2}\qquad \mathrm{(E) \ }  \frac{2}{3} </math>
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<math> \textbf{(A) } \frac{1}{5}\qquad \textbf{(B) }  \frac{1}{4}\qquad \textbf{(C) }  \frac{1}{3}\qquad \textbf{(D) }  \frac{1}{2}\qquad \textbf{(E) }  \frac{2}{3} </math>
  
 
==Solution==
 
==Solution==
 
There are at most <math>5</math> games played.  
 
There are at most <math>5</math> games played.  
  
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If team B won the first two games, team A would need to win the next three games. So the only possible order of wins is BBAAA.  
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If team <math>B</math> won the first two games, team <math>A</math> would need to win the next three games. So the only possible order of wins is <math>BBAAA</math>.  
  
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If team A won the first game, and team B won the second game, the possible order of wins are: ABBAA, ABABA, and ABAAX, where X denotes that the 5th game wasn't played.  
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If team <math>A</math> won the first game, and team <math>B</math> won the second game, the possible order of wins are: <math>ABBAA, ABABA,</math> and <math>ABAAX</math>, where <math>X</math> denotes that the <math>5</math>th game wasn't played.  
  
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There is <math>1</math> possibility where team B wins the first game and <math>4</math> total possibilities when team A wins the tournament and team B wins the second game. However, note that the fourth possibility (ABAAX) just as often as the others, so we put <math>1</math> over <math>4</math> total possibilities. The desired probability is then <math>\frac{1}{4}\Rightarrow \boxed{A}.</math>
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There is <math>1</math> possibility where team <math>B</math> wins the first game and <math>4</math> total possibilities when team <math>A</math> wins the series and team <math>B</math> wins the second game. Note that the fourth possibility <math>(ABAAX)</math> occurs twice as often as the others, so we put <math>1</math> over <math>5</math> total possibilities. The desired probability is then <math>\boxed{\textbf{(A) }\frac{1}{5}}</math>.
  
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In actuality, the fourth possibility (ABAAX) simply counts as 1 case, and it does not occur twice as often as the others because we are given the information that team A will win by the problem.  
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<math>ABAAX</math> counts as two cases because it can be either <math>ABAAB</math> or <math>ABAAA</math>.
  
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We have 4 cases, out of which only 1 (BBAAA) is desired. Thus, our answer is <math>\frac{1}{4}\Rightarrow \boxed{B}.</math> -Flames
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~lnzhonglp
  
 
==See Also==
 
==See Also==
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{{AMC10 box|year=2005|ab=A|num-b=17|num-a=19}}
 
{{AMC10 box|year=2005|ab=A|num-b=17|num-a=19}}
  
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[[Category:Introductory Geometry Problems]]
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[[Category:Introductory Combinatorics Problems]]
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[[Category:Area Ratio Problems]]
 
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 12:38, 10 August 2022

Problem

Team A and team B play a series. The first team to win three games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team B wins the second game and team A wins the series, what is the probability that team B wins the first game?

$\textbf{(A) } \frac{1}{5}\qquad \textbf{(B) } \frac{1}{4}\qquad \textbf{(C) } \frac{1}{3}\qquad \textbf{(D) } \frac{1}{2}\qquad \textbf{(E) } \frac{2}{3}$

Solution

There are at most $5$ games played.

If team $B$ won the first two games, team $A$ would need to win the next three games. So the only possible order of wins is $BBAAA$.

If team $A$ won the first game, and team $B$ won the second game, the possible order of wins are: $ABBAA, ABABA,$ and $ABAAX$, where $X$ denotes that the $5$th game wasn't played.

There is $1$ possibility where team $B$ wins the first game and $4$ total possibilities when team $A$ wins the series and team $B$ wins the second game. Note that the fourth possibility $(ABAAX)$ occurs twice as often as the others, so we put $1$ over $5$ total possibilities. The desired probability is then $\boxed{\textbf{(A) }\frac{1}{5}}$.

$ABAAX$ counts as two cases because it can be either $ABAAB$ or $ABAAA$.

~lnzhonglp

See Also

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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