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    Multinomial Distribution

    A generalization of the Binomial distribution for more than two outcomes.

    Beyond Success or Failure

    The Multinomial distribution extends the Binomial distribution to situations with more than two possible outcomes for each trial. While Binomial models the number of successes in a series of 'success/failure' trials, Multinomial models the number of times each of a set of possible outcomes occurs.

    For example, instead of just a 'win' or 'loss', a trade could result in a 'big win', 'small win', 'breakeven', 'small loss', or 'big loss'. The Multinomial distribution can calculate the probability of observing a specific count for each of these categories over a series of trades.

    The Formula
    The probability of observing a specific set of counts is:
    P(X1=x1,...,Xc=xc)=n!x1!...xc!p1x1⋯pcxcP(X_1=x_1, ..., X_c=x_c) = \frac{n!}{x_1!...x_c!} p_1^{x_1} \cdots p_c^{x_c}P(X1​=x1​,...,Xc​=xc​)=x1​!...xc​!n!​p1x1​​⋯pcxc​​
    • nnn is the total number of trials.
    • ccc is the number of possible outcomes.
    • xix_ixi​ is the number of times outcome iii occurred.
    • pip_ipi​ is the probability of outcome iii on a single trial.
    Interactive Multinomial Calculator
    Specify the parameters to calculate the probability of a specific result.
    OutcomeProbability (pip_ipi​)Desired Count (xix_ixi​)
    Win
    Loss
    Draw