Advanced Counting and Probability Math Club Insights
Explore advanced topics in counting and probability, including choosing from sets, the inclusion-exclusion principle, poker probabilities, and more. Learn how to calculate probabilities in poker, solve project Euler problems, and tackle challenging AIME questions.
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Presentation Transcript
Advanced (ish) Counting and Probability Math Club 10/31/2011 Page 1
Choosing from a set ? ? The symbol there are to choose ? things out of a set of ? things. denotes the number of ways ? ?=? ? ? (? ?+?) ? ? ? ? ?=? ? For instance, there are choose 2 pumpkins out of a set of 5 pumpkins. ways to ? ?= ?? Page 2
Inclusion Exclusion Principle What if you need to count the number of things satisfying property A or B? If you just add up the number of things of property A and of property overcount. B, you might ? ? ?? ? = ? ? + ? ? ?(? ??? ?) Page 3
Inclusion Exclusion Principle What if you want to count things that satisfy at least one of any of the properties A, B, or C? ? ? ?? ? ?? ? = ? ? + ? ? + ? ? ? ? ??? ? + ? ? ??? ? + ? ? ??? ? + ?(? ??? ? ??? ?) Page 4
Inclusion Exclusion Principle: Pattern?! What if you want to count things satisfying ?? or ??or or ??? ? ?? + + ? ?? ? ??????? ? ??????? +? ???????????? + ? ????? ????? ?? + ???(n n things things here here) Page 5
Easy Problem (Poker Probabilities) A set of five playing cards is considered a full house if it contains three cards of one rank and two cards of another rank. What is the probability that five random cards is a full house? Page 6
Easy Problem (Project Euler 1) What is the sum of all the positive integers below 1000 that are divisible by 3 or 5? Page 7
Easy Problem (AIME 2011 II #12) Nine delegates per country) sit at a round table. What s the probability that everyone sits next to at least one person from a different country? delegates from three countries (three Page 8
Problem of The Day I take two shuffled decks of 52 cards each, and for 52 turns deal one card simultaneously. What s the chance that at some point the two cards are the same? from each deck Page 9
