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Contemporary Mathematics

Formula Review

Contemporary MathematicsFormula Review

Formula Review

7.2 Permutations

  • Pnr=n!(n−r)!Pnr=n!(n−r)!

7.3 Combinations

  • The formula for counting combinations is:
     nCr=n!r!(n−r)!nCr=n!r!(n−r)!

7.5 Basic Concepts of Probability

  • For an experiment whose sample space SS consists of equally likely outcomes, the theoretical probability of the event EE is the ratio P(E)=n(E)n(S)P(E)=n(E)n(S) where n(E)n(E) and n(S)n(S) denote the number of outcomes in the event and in the sample space, respectively.
  • P(E)=1−P(E′)P(E)=1−P(E′)

7.7 What Are the Odds?

  • For an event EE,
    odds forE=n(E):n(E′)=P(E):P(E′)=P(E):(1−P(E))odds againstE=n(E′):n(E)=P(E′):P(E)=(1−P(E)):P(E)odds forE=n(E):n(E′)=P(E):P(E′)=P(E):(1−P(E))odds againstE=n(E′):n(E)=P(E′):P(E)=(1−P(E)):P(E)
  • If the odds in favor of EE are A:BA:B, then P(E)=AA+BP(E)=AA+B.

7.8 The Addition Rule for Probability

  • If EE and FF are mutually exclusive events, then
    P(EorF)=P(E)+P(F)P(EorF)=P(E)+P(F).
  • If EE and FF are events that contain outcomes of a single experiment, then
    P(EorF)=P(E)+P(F)−P(E⁢and⁢F)P(EorF)=P(E)+P(F)−P(E⁢and⁢F).

7.9 Conditional Probability and the Multiplication Rule

  • If EE and FF are events associated with the first and second stages of an experiment, then P(Eand⁢F)=P(E)×P(F|E)P(Eand⁢F)=P(E)×P(F|E).

7.10 The Binomial Distribution

  • Suppose we have a binomial experiment with nn trials and the probability of success in each trial is pp. Then:
     P(number of successes isa)=Cna×pa×(1−p)n−aP(number of successes isa)=Cna×pa×(1−p)n−a

7.11 Expected Value

  • If OO represents an outcome of an experiment and n(O)n(O) represents the value of that outcome, then the expected value of the experiment is:
    ∑ n(O)×P(O)∑ n(O)×P(O)

    where ΣΣ stands for the sum, meaning we add up the results of the formula that follows over all possible outcomes.

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