chore: add balls and bins simulator (#2001)

* chore: add balls and bins simulator

Signed-off-by: Roman Gershman <roman@dragonflydb.io>

* Update balls_bins.py

Signed-off-by: Roman Gershman <roman@dragonflydb.io>

---------

Signed-off-by: Roman Gershman <roman@dragonflydb.io>

tools/balls_bins.py

@@ -0,0 +1,72 @@
+#!/usr/bin/env python3
+
+"""Simulate throwing balls into bins."""
+
+import numpy as np
+import argparse
+import matplotlib.pyplot as plt
+
+
+def simulate_balls_into_bins(balls: int, bins: int, threshold: int, exact, trials=10000):
+    """Simulate throwing M balls into N bins for a given number of trials."""
+    counts = np.zeros(bins, dtype=int)
+    success = 0
+    exact_success = 0
+    deltas = []
+
+    for _ in range(trials):
+        # Reset counts for each trial
+        counts.fill(0)
+
+        # Throw M balls into the bins
+        bins_seq = np.random.randint(0, bins, balls)
+        unique, counts_bins = np.unique(bins_seq, return_counts=True)
+        counts[unique] += counts_bins
+        deltas.append(counts.max() - counts.min())
+        # Check if any bin has K or more balls
+        if np.any(counts >= threshold):
+            success += 1
+        if exact is not None:
+            if np.any(counts == exact):
+                exact_success += 1
+
+    probability = success / trials
+    return deltas, probability, exact_success / trials
+
+
+def main():
+    parser = argparse.ArgumentParser(description="Simulate throwing balls into bins.")
+    parser.add_argument("--balls", type=int, default=30, help="Number of balls to throw.")
+    parser.add_argument("--bins", type=int, default=3, help="Number of bins.")
+    parser.add_argument(
+        "--high-threshold",
+        type=int,
+        default=15,
+        help="Minimum number of balls for the success condition",
+    )
+    parser.add_argument(
+        "--exact-num", type=int, help="Exact number of balls for the success condition."
+    )
+    parser.add_argument(
+        "--trials", type=int, default=10000, help="Number of trials. Default is 10,000."
+    )
+
+    args = parser.parse_args()
+
+    deltas, atleast_p, exact_p = simulate_balls_into_bins(
+        args.balls, args.bins, args.high_threshold, args.exact_num, args.trials
+    )
+
+    print(f"Probability that at least one bin has {args.high_threshold} or more balls: {atleast_p}")
+    if args.exact_num is not None:
+        print(f"Probability that at least one bin has {args.exact_num} balls: {exact_p}")
+
+    print(
+        f"Histogram of the difference between the most and least populated bins for {args.trials} trials"
+    )
+    plt.hist(deltas, bins=30, color="steelblue", edgecolor="none")
+    plt.show()
+
+
+if __name__ == "__main__":
+    main()