The Law of Large Numbers

Part of the Statistics for ML section of Coddy's Machine Learning journey. Lesson 14 of 64.

Toss a coin 10 times and 8 heads is no surprise. Keep tossing, and the share of heads crawls toward 0.5:

A line chart of the share of heads over 300 tosses: it swings between 0.3 and 1 in the first tosses, then hugs a dashed red line at the probability 0.5

Each point needs the share so far. np.cumsum gives the running total; divide it by the tosses so far:

heads = np.array([True, False, True, True])
print(np.cumsum(heads))
print(np.round(np.cumsum(heads) / np.arange(1, 5), 2))

This prints:

[1 1 2 3]
[1.   0.5  0.67 0.75]

ax.axhline(y) draws the reference line straight across the chart:

ax.plot(np.arange(1, 301), share)
ax.axhline(0.5, color="red", linestyle="--")

That is the law of large numbers: more tries, closer to the probability. A model scoring well on 20 samples may be lucky; on 2000, hardly.

Challenge

Medium

A cereal brand hides a toy in 20% of boxes. The program reads how many get opened; the starter fills toys (True for a toy) and labels the chart.

Compute the running share of toys.
Print it after 10 boxes and after all, to 3 decimals.
Plot it against box numbers from 1, add a black dotted line (":") at 0.2 and show it.

For 80 other boxes:

After 10 boxes: 0.1
After 80 boxes: 0.212

and draws this chart. Then Coddy describes yours.

A chart titled Toys in Cereal Boxes: the running share over 80 boxes sits at 0 for 8 boxes, jumps around, then settles near a black dotted line at 0.2

Try it yourself

import numpy as np
import matplotlib.pyplot as plt

boxes = int(input())

# Don't change the lines below
rng = np.random.default_rng(25)
toys = rng.random(boxes) < 0.2

fig, ax = plt.subplots()
ax.set_title("Toys in Cereal Boxes")
ax.set_xlabel("Boxes opened")
ax.set_ylabel("Share with a toy")

# Write your code below

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Statistics for ML

3Probability by Simulation

Probability as FrequencyThe Law of Large NumbersSums of Dice Expected Value Conditional Probability Recap: Dice Stall Odds

6Relationships

Reading Scatter Plots Correlation Correlation Matrix Correlation Heatmaps Correlation Is Not Cause Recap: Ice Cream and Sun

9Feature Scaling

Why Scale Features Min-Max Scaling Standardization StandardScaler Recap: Climbing Partners

12Final Challenges

Train Delays Air Quality Wine Chemistry Clinic Survey

Practice on your own: Online Python compiler