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  1. \include{mathcounts_chicken_00}
  2.  
  3. \begin{document}
  4. \title{\textit{MathCounts-AMC8 Year of the Chicken\\Winter-5, Combinatorics-2}}
  5. \maketitle
  6. \thispagestyle{fancy}
  7. \section{Lecture Problems}
  8. \begin{enumerate}
  9. %topics: probability
  10. \pr{}{}{(MathCounts-2009-State-Sprint-06)}{The probability it will rain on Saturday is 60\%, and the probability it will rain on Sunday is 25\%. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent?}{15}{Rick Huang}{Since the probability of raining on Saturday and Sunday is 0.6 and 0.25, respectively, then we simply multiply them together as they are independent events to get $0.6 \times 0.25$ = $0.15$ or 15\% of raining on both days.}
  11.  
  12. \pr{}{}{(MathCounts-2009-School-Sprint-06)}{Jamie rolled a standard six-sided die that has each face painted either red, green or blue. In 100 rolls, the die landed with a red face up 50 times, with a blue face up 17 times, and with a green face up 33 times. Based on this data, how many faces of the die would you expect to be green?}{2}{Rick Huang}{The number of times green was rolled is $\frac{33}{100}$, which corresponds to approximately $\frac{1}{3}$ of the time. Since there are six faces on a die, it's expected that $\frac{1}{3}$ or 2 faces are green.}
  13.  
  14. \pr{}{}{(Canada-Gauss8-2009-09)}{If Jeff picks one letter randomly from the alphabet, what is the probability that the
  15. letter is in the word `probability'?\\
  16. \\(A) 9/26 \quad(B) 15/26 \quad(C) 10/26 \quad(D) 17/26 \quad(E) 8/26}{A}{Rick Huang}{`Probability' has nine different letters, namely `p,' `r,' `o,' `b,' `a,' `i,' `l,' `t,' and `y.' Therefore, we can pick one of these nine letters of the twenty-six total letters within the alphabet and have it be part of the word `probability,' giving a $\frac{9}{26}$ chance, or A.}
  17.  
  18. \pr{}{}{(MathCounts-2009-Chapter-Sprint-10)}{Rayna writes each letter of her name—R, A, Y, N and A—on a separate piece of 1-by-1 paper. She then puts the 5 pieces of paper in a bag and randomly selects one of the 5 pieces. What is the probability that she selects the piece with the R written on it? Express your answer as a common fraction.}{1/5}{Rick Huang}{There is one `R' out of five total pieces of paper, so the probability is simply $\frac{1}{5}$.}
  19.  
  20. \pr{}{}{(AMC8-2009-10)}{On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?\\
  21. \includegraphics[width=2in]{images/AMC8-2009-10.png}
  22. \\(A) 1/16 \quad(B) 7/16 \quad(C) 1/2 \quad(D) 9/16 \quad(E) 49/64}{D}{}{}
  23.  
  24. \pr{}{}{(AMC8-2011-12)}{Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other? \\
  25. \\(A) 1/4 \quad(B) 1/3 \quad(C) 1/2 \quad(D) 2/3 \quad(E) 3/4}{B}{}{}
  26.  
  27. \pr{}{}{(MathCounts-2009-State-Sprint-14)}{If three standard, six-faced dice are rolled, what is the probability that the sum of the three numbers rolled is 9? Express your answer as a common fraction.}{25/216}{}{}
  28.  
  29. \pr{}{}{(MathCounts-2001-Chapter-Sprint-18)}{Two circles with spinners at their centers are divided into three equal regions as shown. When both spinners are spun, what is the probability that the product of the two values is negative? Express your answer as a common fraction.\\
  30. \includegraphics[width=2in]{images/MathCounts-2001-Chapter-Sprint-18.png}}{5/9}{}{}
  31.  
  32. \pr{}{}{(AMC8-2011-18)}{A fair 6-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?  \\
  33. \\(A) 1/6 \quad(B) 5/12 \quad(C) 1/2 \quad(D) 7/12 \quad(E) 5/6}{D}{}{}
  34.  
  35. \pr{}{probability;}{(AMC10A-2005-09)}{Three tiles are marked $X$ and two other tiles are marked $O$. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads $XOXOX$?\\
  36. \\(A) $\dfrac{1}{12}$  \quad(B) $\dfrac{1}{10}$  \quad(C) $\dfrac{1}{6}$  \quad(D) $\dfrac{1}{4}$  \quad(E) $\dfrac{1}{3}$}{B}{}{}
  37.  
  38. \pr{}{}{(AMC8-2007-21)}{Two cards are dealt from a deck of four red cards labeled $A,B,C,D$ and four
  39. green cards labeled $A,B,C,D$. A winning pair is two of the same color or two
  40. of the same letter. What is the probability of drawing a winning pair?\\
  41. \\(A) 2/7 \quad(B) 3/8 \quad(C) 1/2 \quad(D) 4/7 \quad(E) 5/8}{D}{}{}
  42.  
  43. \pr{}{}{(AMC12-2001-11)}{A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?\\\\
  44. $\text{(A) }\dfrac {3}{10}\qquad\text{(B) }\dfrac {2}{5}\qquad\text{(C) }\dfrac {1}{2}\qquad\text{(D) }\dfrac {3}{5}\qquad\text{(E)}\dfrac {7}{10}$}{D}{}{}
  45.  
  46. \pr{}{}{(MathCounts-2007-State-Sprint-23)}{A box contains some green marbles and exactly four red marbles.
  47. The probability of selecting a red marble is $x\%$. If the number of
  48. green marbles is doubled, the probability of selecting one of the
  49. four red marbles from the box is $(x - 15)\%$. How many green
  50. marbles are in the box before the number of green marbles is
  51. doubled?}{6}{}{}
  52.  
  53. \pr{}{}{(MathCounts-2005-State-Sprint-24)}{There are only red marbles and green marbles in a bag. The ratio of red marbles to green marbles in the bag is 4:7. Julia then adds 90 red marbles and 36 green marbles to the bag, which makes the probability of selecting a red marble from the bag on a random draw equal to 1/2. How many total marbles are in the bag after Julia has added the 126 marbles?}{324}{}{}
  54.  
  55. \pr{}{}{(AMC12A-2007-12)}{Integers $a, b, c$, and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?\\
  56. \\(A) 3/8 \quad(B) 7/16 \quad(C) 1/2 \quad(D) 9/16 \quad(E) 5/8}{E}{}{}
  57.  
  58. \pr{}{}{(AMC12B-2007-13)}{A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?\\\\
  59. $\mathrm {(A)} \dfrac{1}{63} \qquad \mathrm {(B)} \dfrac{1}{21} \qquad \mathrm {(C)} \dfrac{1}{10} \qquad \mathrm {(D)} \dfrac{1}{7} \qquad \mathrm {(E)} \dfrac{1}{3}$}{D}{}{}
  60.  
  61. \pr{}{}{(MathCounts-2011-State-Sprint-27)}{In three flips of an unfair coin the probability of getting three
  62. heads is the same as the probability of getting exactly two
  63. tails. What is the ratio of the probability of flipping a tail to
  64. the probability of flipping a head? Express your answer as a
  65. common fraction in simplest radical form.}{$\dfrac{\sqrt{3}}{3}$}{}{}
  66.  
  67. \pr{}{}{(MathCounts-2007-Chapter-Sprint-28)}{Four couples are at a party. Four people of the eight are
  68. randomly selected to win a prize. No person can win more
  69. than one prize. What is the probability that both members
  70. of at least one couple win a prize? Express your answer as a
  71. common fraction.}{27/35}{}{}
  72.  
  73. \pr{}{probability;}{(AMC12A-2005-14)}{On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? \\
  74. \\(A) 5/11  \quad(B) 10/21  \quad(C) 1/2  \quad(D) 11/21  \quad(E) 6/11}{D}{}{}
  75.  
  76. \pr{}{}{(MathCounts-2011-State-Sprint-29)}{A bag contains red balls and white balls. If five balls
  77. are to be pulled from the bag, with replacement, the
  78. probability of getting exactly three red balls is 32 times the
  79. probability of getting exactly one red ball. What percent of
  80. the balls originally in the bag are red?}{80}{}{}
  81.  
  82. \pr{}{}{(AMC10A-2003-15)}{What is the probability that an integer in the set $\{1,2,3,\ldots,100\}$ is divisible by $2$ and not divisible by $3$?\\\\
  83. $ \mathrm{(A) \ } \dfrac{1}{6}\qquad \mathrm{(B) \ }  \dfrac{33}{100}\qquad \mathrm{(C) \ }  \dfrac{17}{50}\qquad \mathrm{(D) \ }  \dfrac{1}{2}\qquad \mathrm{(E) \ }  \dfrac{18}{25} $}{C}{}{}
  84.  
  85. \pr{}{}{(MathCounts-2007-School-Target-2)}{In Sumville, 40\% of the citizens are at least 6 feet tall and 25\% of the citizens have red hair. Height and hair color have no correlation; the people with red hair are proportionally distributed among the people at least 6 feet tall and the people less than 6 feet tall. Sumville has exactly 500 citizens. How many citizens of Sumville have red hair and are less than 6 feet tall?}{75}{}{}
  86.  
  87. \pr{}{}{(MathCounts-2011-School-Target-2)}{If this shape is folded into a cube and
  88. rolled, one face of the cube will be on
  89. the table while the other faces will be
  90. showing. What is the probability that
  91. the sum of the numbers on the faces that
  92. are showing will be odd? Express your
  93. answer as a common fraction.\\
  94. \includegraphics[width=1.5in]{images/MathCounts-2011-School-Target-2.png}}{1/6}{}{}
  95.  
  96. \pr{}{}{(MathCounts-2001-State-Target-4)}{In a single elimination tournament, the better player always won. The winner of Round 3 was the champion, and the loser of Round 3 was the runner-up. Eight players are randomly assigned slots in Round 1. What is the probability that the runner-up was not the second-best player in the tournament? Express your answer as a common fraction.\\
  97. \includegraphics[width=3in]{images/MathCounts-2001-State-Target-4.png}}{3/7}{}{}
  98.  
  99. \pr{}{}{(MathCounts-2003-State-Target-6)}{Paco uses a spinner to select a number from 1 through 5, each
  100. with equal probability. Manu uses a different spinner to select
  101. a number from 1 through 10, each with equal probability.
  102. What is the probability that the product of Manu's number and
  103. Paco's number is less than 30? Express your answer as a
  104. common fraction.}{41/50}{}{}
  105.  
  106. \pr{}{}{(MathCounts-2011-State-Target-7)}{Hans and Franz are in a shooting competition. The object of the
  107. match is to be the first to hit the bull’s-eye of a target 100 feet
  108. away. The two opponents alternate turns shooting, and each
  109. opponent has a 40\% chance of hitting the bull’s-eye on a given
  110. shot. If Hans graciously allows Franz to shoot first, what is
  111. the probability that Hans will win the competition and take no
  112. more than three shots? Express your answer as a decimal to the
  113. nearest hundredth.}{0.36}{}{}
  114.  
  115. \pr{}{}{(MathCounts-2001-Chapter-Team-01)}{A point $(x,y)$ is randomly selected such that $0\leq x\leq 8$ and $0\leq y\leq 4$. What is the probability that both $x\leq 2$ and $y\leq 2$? Express your answer as a common fraction.}{1/8}{}{}
  116.  
  117. \pr{}{}{(MathCounts-2007-School-Team-06)}{The Charleston Dodgeball Team has 16 players at a tournament match. If each player is equally likely to be chosen as one of the 10 starters, what is the probability that the Charleston player Joe ``The Arm'' Morez will start? Express your answer as a common fraction.}{5/8}{}{}
  118.  
  119. \pr{}{}{(MathCounts-2005-Chapter-Team-09)}{If Ella rolls a standard six-sided die until she rolls the same number on consecutive rolls, what is the probability that her 10th roll is her last roll? Express your answer as a decimal to the nearest thousandth.}{0.039}{}{}
  120.  
  121. \pr{}{}{(MathCounts-2003-Chapter-Team-10)}{An octahedron consists of two square-based pyramids glued
  122. together along their square bases to form a polyhedron
  123. with eight faces. Imagine an ant that begins at the
  124. top vertex and walks to one of the four adjacent
  125. vertices that he randomly selects and calls vertex
  126. A. From vertex A, he will then walk to one of
  127. the four adjacent vertices that he randomly selects
  128. and calls vertex B. What is the probability that vertex
  129. B will be the bottom vertex? Express your answer as a
  130. common fraction.\\
  131. \includegraphics[width=2in]{images/MathCounts-2003-Chapter-Team-10.png}}{1/4}{}{}
  132. \end{enumerate}
  133.  
  134. \newpage
  135. \section{Homework Set}
  136. \begin{enumerate}
  137. \pr{}{}{(MathCounts-2005-State-Sprint-05)}{The chart below gives the air distance in miles between selected world cities. If two different cities from the chart are chosen at random, what is the probability that the distance between them is less than 7000 miles? Express your answer as a common fraction.\\
  138. \includegraphics[width=4.5in]{images/MathCounts-2005-State-Sprint-05.png}}{2/3}{}{}
  139.  
  140. \pr{}{}{(MathCounts-2003-Chapter-Sprint-06)}{The numbers 1 through 25 are written on 25 cards with one
  141. number on each card. Sara picks one of the 25 cards at random.
  142. What is the probability that the number on her card will be a
  143. multiple of 2 or 5? Express your answer as a common fraction.}{3/5}{}{}
  144.  
  145. \pr{}{}{(AMC8-1999-10)}{A complete cycle of a traffic light takes 60 seconds. During each cycle the light is
  146. green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. At a randomly
  147. chosen time, what is the probability that the light will NOT be green?\\
  148. \\(A) 1/4 \quad(B) 1/3 \quad(C) 5/12 \quad(D) 1/2 \quad(E) 7/12}{E}{}{}
  149.  
  150. \pr{}{}{(AMC8-2009-12)}{The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?\\
  151. \includegraphics[width=2in]{images/AMC8-2009-12.png}
  152. \\(A) 1/2 \quad(B) 2/3 \quad(C) 3/4 \quad(D) 7/9 \quad(E) 5/6}{D}{}{}
  153.  
  154. \pr{}{}{(AMC8-2003-12)}{When a fair six-sided die is tossed on a table top, the bottom face cannot
  155. be seen. What is the probability that the product of the numbers on the
  156. five faces that can be seen is divisible by 6?\\
  157. \\(A) 1/3 \quad(B) 1/2 \quad(C) 2/3 \quad(D) 5/6 \quad(E) 1}{E}{}{}
  158.  
  159. \pr{}{}{(AMC8-2009-13)}{A three-digit integer contains one of each of the digits 1, 3, and 5. What is the probability that the integer is divisible by 5?\\
  160. \\(A) 1/6 \quad(B) 1/3 \quad(C) 1/2 \quad(D) 2/3 \quad(E) 5/6}{B}{}{}
  161.  
  162. \pr{}{}{(AMC8-2001-18)}{Two dice are thrown. What is the probability that the product of the
  163. two numbers is a multiple of 5?\\
  164. \\(A) 1/36 \quad(B) 1/18 \quad(C) 1/6 \quad(D) 11/36 \quad(E) 1/3}{D}{}{}
  165.  
  166. \pr{}{}{(MathCounts-2001-State-Sprint-21)}{A standard die is tossed twice. What is the probability of obtaining exactly one 5? Express your answer as a common fraction.}{5/18}{}{}
  167.  
  168. \pr{}{}{(MathCounts-2003-State-Sprint-21)}{Container I holds 8 red balls and 4 green balls; containers II
  169. and III each hold 2 red balls and 4 green balls. A container is
  170. selected at random and then a ball is randomly selected from
  171. that container. What is the probability that the ball selected is
  172. green? Express your answer as a common fraction.}{5/9}{}{}
  173.  
  174. \pr{}{}{(MathCounts-2011-School-Sprint-22)}{Alice rolls a standard 6-sided die once. Bob rolls a second
  175. standard 6-sided die once. Alice wins if the values shown have
  176. a positive difference of exactly 1. What is the probability that
  177. Alice wins? Express your answer as a common fraction.}{5/18}{}{}
  178.  
  179. \pr{}{}{(AMC8-2007-24)}{A bag contains four pieces of paper, each labeled with one of the digits 1, 2, 3
  180. or 4, with no repeats. Three of these pieces are drawn, one at a time without
  181. replacement, to construct a three-digit number. What is the probability that
  182. the three-digit number is a multiple of 3?\\
  183. \\(A) 1/4 \quad(B) 1/3 \quad(C) 1/2 \quad(D) 2/3 \quad(E) 3/4}{C}{}{}
  184.  
  185. \pr{}{}{(MathCounts-2011-Chapter-Sprint-26)}{If a committee of six students is chosen at random from a group of six boys and four girls, what is the probability that the committee contains the same number of boys and girls? Express your answer as a common fraction.}{8/21}{}{}
  186.  
  187. \pr{}{probability;}{(AMC10B-2011-13)}{Two real numbers are selected independently at random from the interval $[-20,10]$. What is the probability that the product of those numbers is greater than zero? \\
  188. \\(A) 1/9  \quad(B) 1/3  \quad(C) 4/9  \quad(D) 5/9  \quad(E) 2/3}{D}{}{}
  189.  
  190. \pr{}{}{(MathCounts-2003-State-Sprint-28)}{Two different numbers are randomly selected from the set
  191. $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. The probability that their
  192. sum is 12 would be greater if the number n had first been
  193. removed from set $S$. What is the value of $n$?}{6}{}{}
  194.  
  195. \pr{}{}{(AMC10A-2011-14)}{A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? \\
  196. \\(A) 1/36  \quad(B) 1/12  \quad(C) 1/6  \quad(D) 1/4  \quad(E) 5/18}{B}{}{}
  197.  
  198. \pr{}{}{(MathCounts-2009-Chapter-Sprint-29)}{In a 7-by-7 checkerboard, two unit squares will be chosen at random and without replacement. What is the probability that the two squares are adjacent to each other (share a side)? Express your answer as a common fraction.}{1/14}{}{}
  199.  
  200. \pr{}{}{(MathCounts-2003-Chapter-Sprint-29)}{Each day, two out of the three teams in a class are randomly
  201. selected to participate in a MATHCOUNTS trial competition.
  202. What is the probability that Team A is selected on at least two
  203. of the next three days? Express your answer as a common
  204. fraction.}{20/27}{}{}
  205.  
  206. \pr{}{}{(AMC12B-2003-19)}{Let $S$ be the set of permutations of the sequence $1,\ 2,\ 3,\ 4,\ 5$ for which the first term is not $1$. A permutation is chosen randomly from $S$. The probability that the second term is $2$, in lowest terms, is $a/b$. What is $a+b$?\\\\
  207. $\mathrm{(A)}\ 5 \qquad\mathrm{(B)}\ 6 \qquad\mathrm{(C)}\ 11 \qquad\mathrm{(D)}\ 16 \qquad\mathrm{(E)}\ 19$}{E}{}{}
  208.  
  209. \pr{}{}{(MathCounts-2009-Chapter-Target-5)}{When rolling two standard sixsided dice, what is the probability of getting a sum larger than 10? Express your answer as a common fraction.}{1/12}{}{}
  210.  
  211. \pr{}{}{(MathCounts-2007-Chapter-Target-7)}{Six boys and six girls are seated randomly in a row of
  212. 12 chairs. What is the probability that no two boys are seated
  213. next to one another and no two girls are seated next to one
  214. another? Express your answer as a common fraction.}{1/462}{}{}
  215.  
  216. \pr{}{}{(MathCounts-2003-Chapter-Target-7)}{Two distinct numbers are selected simultaneously and at
  217. random from the set $\{1, 2, 3, 4, 5\}$. What is the probability that
  218. their product is an even number? Express your answer as a
  219. common fraction.}{7/10}{}{}
  220.  
  221. \pr{}{}{(MathCounts-2001-State-Team-05)}{A three-digit number is created with three different digits from the set $\{1,2,3,4,5\}$. What is the probability that the number is a multiple of 15? Express your answer as a common fraction?}{1/15}{}{}
  222.  
  223. \pr{}{}{(MathCounts-2011-State-Team-09)}{A bag contains five red marbles, three blue marbles and two
  224. green marbles. Six marbles are to be drawn from the bag,
  225. replacing each one after it is drawn. What is the probability that
  226. two marbles of each color will be drawn? Express your answer
  227. as a common fraction.}{81/1000}{}{}
  228.  
  229. \pr{}{}{(MathCounts-2009-Chapter-Team-10)}{A bag contains red marbles, white marbles, green marbles and blue marbles. There are an equal number of red marbles and white marbles, and five times as many green marbles as blue marbles. There is a 35\% chance of selecting a red marble first. What is the fewest possible number of green marbles in the bag?}{5}{}{}
  230. \end{enumerate}
  231. \end{document}
  232.  
  233.  
  234.  
  235. ------------------------------------------------------------------------------------------------------------------------------------------------
  236.  
  237. \pr{}{}{(MathCounts-2003-State-Team-07)}{$P(n)$ represents the probability that an ``$n$'' is rolled on a die. A
  238. six-faced die, with faces labeled 1 through 6, is weighted such that:\\
  239. $\bullet$ $P(1)=P(2)$\\
  240. $\bullet$ $P(3)=P(4)=P(5)$\\
  241. $\bullet$ $P(4)=3P(2)$\\
  242. $\bullet$ $P(5)=2P(6)$.\\
  243. If this die is rolled once, what is the probability that a ``6'' is rolled? Express your answer as a common fraction.}{3/25}{}{}
  244.  
  245. \pr{}{}{(MathCounts-2007-State-Team-10)}{In a bag of three marbles, there are exactly two blue marbles.
  246. If Kia randomly chooses two marbles without replacement,
  247. the probability of choosing the two blue marbles is one-third.
  248. However, before Kia chooses her two marbles, additional
  249. marbles are added to the bag. The probability of picking two
  250. blue marbles without replacement is still one-third. What is
  251. the least number of marbles that could be in the bag after the
  252. additional marbles have been added?}{10}{}{}
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