def false_negative_rate_binary(num_positives, type='exp'):
    """
    computes the false negative rate (FNR) of a pooled PCR test, based on the number of
    positive samples in the pooled sample

    INPUT:
    num_positives = number of positive samples in the pooled sample
    type = false negative rate function used in the calculation
    """
    assert num_positives >= 0

    if type == 'exp':
        return 0.1 ** num_positives

    elif type == 'reciprocal':
        return 1 / (1 + num_positives)

    else:
        assert type == 'step'
        return 1 - (num_positives >= 3)