import numpy as np


def apply_randomized_response(bits: np.ndarray, epsilon: float) -> np.ndarray:
    keep_prob = np.exp(epsilon) / (1 + np.exp(epsilon))  # sigmoid fn

    # Generate random decisions for each bit
    keep_mask = np.random.random(bits.shape) < keep_prob

    # For bits we don't keep, flip them
    noisy_bits = np.where(keep_mask, bits, 1 - bits)

    return noisy_bits