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Understanding Probability: Binomial Theorem, Permutations & Combinations Explained

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Dive into the fundamentals of probability with this comprehensive guide covering key concepts such as the Binomial Theorem, permutations, and combinations. Learn how to apply the Fundamental Counting Principle, distinguish between independent and mutually exclusive events, and understand how to calculate outcomes and probabilities. This resource is perfect for students and anyone looking to enhance their understanding of probability theory, providing clear explanations and practical examples for mastering these essential topics.

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Understanding Probability: Binomial Theorem, Permutations & Combinations Explained

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  1. 9B Probability Daniel Meissner Nick Lauber Kaitlyn Stangl Lauren Desordi

  2. Vocabulary Binomial Theorem Permutation Combinations Independent Mutually Exclusive

  3. The Binomial Theorem • (a+b)n = nC0anb0 + nC1an-1b1 + nC2an-2b2 +….nCnA0bn

  4. Fundamental Counting Principle If event E1 can occur m1 different ways and event E2 can occur m2 different ways then the number of ways they can both occur is m1 * m2 Equation for total possible outcomes: m1 * m2…. *mk

  5. Permutations An arrangement of objects where order matters n! = Number of permutations of n objects nPr = Number of permutations of n objects taken r at a time

  6. Distinguishable Permutations • If a set of n objects has n1 of one kind, n2 of another kind etc… The number of distinguishable permutations

  7. Combinations An arrangement where order does not matter nCr: Number of combinations of n objects taken r at a time

  8. Experiment A happening for which the results is uncertain • Outcomes: Possible results • Sample Space: The set of all possible outcomes • Event: A subset of the sample space

  9. Probability If an event E has n(E) equally likely outcomes and its sample space s has s(E) equally likely outcomes then the probability of event E is Compliments: The probability that event E will not happen P(E’) = 1 – P(E)

  10. Compound Probability Events in the same sample space that have no common outcomes: P(A n B) = 0 If A and B are 2 events in the same sample space, then the probability of A or B is P(A u B) = P(A) + P(B) – P(A n B) If A & B are mutually exclusive, then just P(A u B) = P(A) + P(B) Two events are independent if the occurrence of one event has no effect on the occurrence of the other event

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