Difference between revisions of "2006 AMC 10A Problems/Problem 13"

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<math>\mathrm{(A) \ } $12\qquad\mathrm{(B) \ } $30\qquad\mathrm{(C) \ } $50\qquad\mathrm{(D) \ } $60\qquad\mathrm{(E) \ } $100\qquad</math>
 
<math>\mathrm{(A) \ } $12\qquad\mathrm{(B) \ } $30\qquad\mathrm{(C) \ } $50\qquad\mathrm{(D) \ } $60\qquad\mathrm{(E) \ } $100\qquad</math>
 
== Solution ==
 
== Solution ==
There are <math>6 \cdot 6 = 36</math> possible [[combination]]s of 2 dice rolls. The only possible winning combinations are <math> (2,2) </math>, <math>(4,4)</math> and <math> (6,6) </math>. Since there are <math>3</math> winning combinations and <math>36</math> possible combinations of dice rolls, the probability of winning is <math>\frac{3}{36}=\frac{1}{12}</math>.
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The probability of rolling an even number on the first turn is <math>\frac{1}{2}</math> and the probability of rolling the same number on the next turn is <math>\frac{1}{6}</math>, so the probability of winning is <math>\frac{1}{12}</math>. If the game is to be fair, the amount paid, 5 dollars, must be <math>\frac{1}{12}</math> the amount of prize money, so the answer is '''D.''' 60
 
 
Let <math>x</math> be the amount won in a fair game. By the definition of a fair game,
 
 
 
:<math>\frac{1}{12} \cdot x = 5 </math>.
 
 
 
Therefore, <math> x = \$60 \Longrightarrow \mathrm{D} </math>.
 
  
 
== See also ==
 
== See also ==

Revision as of 14:26, 5 February 2012

Problem

A player pays $5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)

$\mathrm{(A) \ } $12\qquad\mathrm{(B) \ } $30\qquad\mathrm{(C) \ } $50\qquad\mathrm{(D) \ } $60\qquad\mathrm{(E) \ } $100\qquad$ (Error compiling LaTeX. ! LaTeX Error: \mathrm allowed only in math mode.)

Solution

The probability of rolling an even number on the first turn is $\frac{1}{2}$ and the probability of rolling the same number on the next turn is $\frac{1}{6}$, so the probability of winning is $\frac{1}{12}$. If the game is to be fair, the amount paid, 5 dollars, must be $\frac{1}{12}$ the amount of prize money, so the answer is D. 60

See also

2006 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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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