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# numpy.random.multinomial

random.multinomial(n, pvals, size=None)

  Draw samples from a multinomial distribution.

  The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of `p` possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, `X_i = [X_0, X_1, ..., X_p]`, represent the number of times the outcome was `i`.

  Note

  New code should use the `multinomial` method of a `Generator` instance instead; please see the Quick start.

  Warning

  This function defaults to the C-long dtype, which is 32bit on windows and otherwise 64bit on 64bit platforms (and 32bit on 32bit ones). Since NumPy 2.0, NumPy’s default integer is 32bit on 32bit platforms and 64bit on 64bit platforms.

  Parameters:
    n int

      Number of experiments.

    pvals sequence of floats, length p

      Probabilities of each of the `p` different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as `sum(pvals[:-1]) <= 1)`.

    size int or tuple of ints, optional

      Output shape. If the given shape is, e.g., `(m, n, k)`, then `m * n * k` samples are drawn. Default is None, in which case a single value is returned.

  Returns:
    out ndarray

      The drawn samples, of shape size, if that was provided. If not, the shape is `(N,)`.

      In other words, each entry `out[i,j,...,:]` is an N-dimensional value drawn from the distribution.

  See also

  `random.Generator.multinomial`

    which should be used for new code.

  Examples

  Throw a dice 20 times:

      >>> np.random.multinomial(20, [1/6.]*6, size=1)
      array([[4, 1, 7, 5, 2, 1]]) # random

  It landed 4 times on 1, once on 2, etc.

  Now, throw the dice 20 times, and 20 times again:

      >>> np.random.multinomial(20, [1/6.]*6, size=2)
      array([[3, 4, 3, 3, 4, 3], # random
             [2, 4, 3, 4, 0, 7]])

  For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

  A loaded die is more likely to land on number 6:

      >>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
      array([11, 16, 14, 17, 16, 26]) # random

  The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

      >>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
      array([38, 62]) # random

  not like:

      >>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
      Traceback (most recent call last):
      ValueError: pvals < 0, pvals > 1 or pvals contains NaNs