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• 2003 AIME II Problems/Problem 13
  ...ed, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is <math>m/n,</ Let <math>P_n</math> represent the probability that the bug is at its starting vertex after <math>n</math> moves. If the b
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2002 AIME II Problems/Problem 12
  ...the ratio of shots made to shots attempted after <math>n</math> shots. The probability that <math>a_{10} = .4</math> and <math>a_n\le.4</math> for all <math>n</ma ...ossible sequence occurring is <math>(.4)^4(.6)^6</math>. Hence the desired probability is
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2001 AIME II Problems/Problem 11
  ...Truncator will win, lose, or tie are each <math>\frac {1}{3}</math>. The [[probability]] that Club Truncator will finish the season with more wins than losses is ...Thus, by the [[complement principle]], the desired probability is half the probability that Club Truncator does not have the same number of wins and losses.
  3 KB (415 words) - 23:25, 20 February 2023
• 2001 AIME II Problems/Problem 9
  ...or red. For each square, either color is equally likely to be used. The [[probability]] of obtaining a grid that does not have a 2-by-2 red square is <math>\frac ...5=417</math> ways to paint the square with the restriction. Therefore, the probability of obtaining a grid that does not have a <math>2 \times 2</math> red square
  8 KB (1,207 words) - 20:04, 5 September 2023
• 2000 AIME II Problems/Problem 3
  ...at these cards are not returned to the deck, let <math>m/n</math> be the [[probability]] that two randomly selected cards also form a pair, where <math>m</math> a [[Category:Intermediate Combinatorics Problems]]
  1 KB (191 words) - 04:27, 4 November 2022
• Subject Classes
  * Intermediate (at the level of hardest [[AMC 12]] problems, the [[AIME]], [[ARML]], and the [[Mandelbrot Competition]]). * [[Introduction to Counting & Probability Course]] [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#be
  2 KB (303 words) - 16:02, 11 July 2006
• Expected value
  A video that goes over the type of Expected value, practical examples, and problems: https://youtu.be/TCFoRx2R2ew ...that particular outcome. If the event <math>Z</math> has a [[continuous]] probability distribution, then <math>E(Z) = \int_z P(z)\cdot z\ dz</math>.
  5 KB (789 words) - 20:56, 10 May 2024
• AoPS Wiki:Resources
  ...ofproblemsolving.com/Forum/viewtopic.php?mode=attach&id=5352 ARML Marathon Problems (PDF)] ...ofproblemsolving.com/Forum/viewtopic.php?mode=attach&id=5353 ARML Marathon Problems (TeX)]
  6 KB (743 words) - 14:37, 20 December 2010
• 2005 Alabama ARML TST Problems/Problem 1
  ...of the other die are <math>1, 2, 2, 3, 3,\text{ and }4</math>. Find the [[probability]] of rolling a sum of <math>9</math> with these two dice. ..., out of a total of <math>6^2</math> possible two-roll combinations, for a probability of <math>\frac 19</math>.
  1 KB (210 words) - 01:30, 3 January 2023
• ITest
  ...est is varied, ranging from simple arithmetic problems to complex Olympiad problems. Winning teams earn recent technology prizes like a video game console of ...ol math concepts, including [[algebra]], [[geometry]], pre-[[calculus]], [[probability]], and [[logic]]. Graphing calculators are allowed, but computers are not
  2 KB (264 words) - 20:31, 13 December 2018
• Parity
  ...asic, checking the parity of numbers is often an useful tactic for solving problems, especially with [[proof by contradiction]]s and [[casework]]. == Problems ==
  4 KB (694 words) - 22:00, 12 January 2024
• 2006 AIME II Problems/Problem 5
  ...obability of obtaining a sum of 7 is <math>47/288</math>. Given that the [[probability]] of obtaining face <math> F </math> is <math> m/n, </math> where <math> m ...be obtained by rolling a 2 and 5, 5 and 2, 3 and 4, or 4 and 3. Each has a probability of <math>\frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}</math>, totaling <mat
  5 KB (712 words) - 12:10, 5 November 2023
• 2006 AIME II Problems/Problem 10
  ...f the tournament, team <math> A </math> beats team <math> B. </math> The [[probability]] that team <math> A </math> finishes with more points than team <math> B < ...r than <math>A</math>. We let this probability be <math>p</math>; then the probability that <math>A</math> and <math>B</math> end with the same score in these fiv
  6 KB (983 words) - 13:42, 8 December 2021
• AoPS Wiki:Article of the Day/Archive
  ...s: if we have an event <math>Z</math> whose outcomes have a [[discrete]] [[probability]] distribution, the... [[Expected value|[more]]] ...etic while presenting a progression of more difficult and more interesting problems that are easily solved using... [[Modular arithmetic/Introduction|[more]]]
  24 KB (3,425 words) - 22:58, 15 February 2008
• 2004 AMC 12B Problems/Problem 23
  ...250,500</math> is a multiple of <math>2004</math>, so there is a very high probability that it is the correct answer. [[Category:Intermediate Algebra Problems]]
  3 KB (533 words) - 14:52, 29 October 2023
• 2005 Alabama ARML TST Problems/Problem 15
  ...positive integers in such a way that when her pair of dice is rolled, the probability of any particular sum occurring is the same as when Virginia rolls her dice Since the probability of any particular sum occurring on Montana's dice is the same as when Virgi
  1 KB (264 words) - 19:43, 19 June 2008
• 2008 AIME I Problems/Problem 9
  ...n of each crate is chosen at random. Let <math>\frac {m}{n}</math> be the probability that the stack of crates is exactly <math>41\mathrm{ft}</math> tall, where ...e heights matter, and each crate is either 3, 4, or 6 feet tall with equal probability. We have the following:
  6 KB (909 words) - 15:39, 8 August 2022
• Mock AIME 1 2007-2008 Problems/Problem 8
  ...f ten <math>0</math>s and/or <math>1</math>s is randomly generated. If the probability that the sequence does not contain two consecutive <math>1</math>s can be w ...eed <math>a_n = F_{n+2}</math>, so <math>a_{10} = F_{12} = 144</math>. The probability is <math>\frac{144}{2^{10}} = \frac{9}{64}</math>, and <math>m+n=\boxed{073
  2 KB (260 words) - 18:14, 2 April 2008
• Category:Intermediate Probability Problems
  This page lists all of the intermediate [[probability]] problems in the [[AoPSWiki]]. [[Category:Probability Problems]]
  23 members (0 subcategories, 0 files) - 20:33, 23 April 2008
• 2008 Mock ARML 1 Problems/Problem 4
  ...rom the hat and replacing it with one of the opposite color. Compute the [[probability]] that, after a sequence of turns, there are <math>5</math> black balls in ...lity of ending with <math>4</math> black balls and a <math>\frac 35</math> probability of ending with <math>2</math> balls. Thus, we have the recursions
  2 KB (242 words) - 09:03, 1 June 2008

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