Law of Large Numbers: Coin Flips Converge to 50%

Animation Specification: Law of Large Numbers with Coin Flips

Animation Description and Purpose

The animation visually demonstrates the Law of Large Numbers by simulating coin flips and showing how the proportion of heads converges to 50% as the number of flips increases. The animation will display a dynamic bar chart updating in real-time, along with a running tally of heads and tails.

Mathematical Elements and Formulas

• Law of Large Numbers: As $n o \infty$, the sample average $\frac{S_n}{n}$ converges to the expected value $\mu$. For a fair coin, $\mu = 0.5$.
• Proportion of Heads: $P_n = \frac{ ext{Number of Heads}}{n}$, where $n$ is the total number of flips.
• Expected Value: $E[X] = 0.5$ for a fair coin flip $X$.

Visual Elements

1. Coin Flip Simulation:

   • A coin (circle with two sides: heads and tails) flipping in the top-left corner.
   • Each flip is animated briefly (0.2 seconds per flip).
   • The coin lands on either heads or tails randomly.

2. Bar Chart:

   • Two vertical bars representing the proportion of heads (left) and tails (right).
   • Heads bar: Blue color (#3498db).
   • Tails bar: Red color (#e74c3c).
   • The bars start at 0 height and grow dynamically as flips occur.
   • The x-axis is labeled "Heads" and "Tails."
   • The y-axis ranges from 0% to 100%.

3. Running Tally:

   • Text display (with opaque background) showing:
     • Total flips: $n$
     • Number of heads: $H$
     • Proportion of heads: $P_n = \frac{H}{n}$
   • Text color: White (#ffffff).
   • Background color: Black (#000000) with 80% opacity.

4. Convergence Line:

   • A horizontal dashed line at $y = 50\%$ to indicate the expected proportion.
   • Line color: Green (#2ecc71).

Animation Timing and Transitions

• Total Duration: 30 seconds.
• Flip Rate: 10 flips per second (300 flips total).
• Bar Update: Bars update smoothly after each flip (0.1-second transition).
• Initial Setup:
  • 0-1 second: Introduce the coin and bars (fade-in).
  • 1-2 seconds: First 10 flips (slower pace for clarity).
• Main Animation:
  • 2-28 seconds: Remaining 290 flips (faster pace).
  • Bars and tally update dynamically.
• Final Emphasis:
  • 28-30 seconds: Highlight the convergence by briefly flashing the 50% line and displaying the final proportion.

Camera Angles and Perspectives

• Static camera angle with a slight zoom-in effect during the final emphasis (28-30 seconds).
• The coin flip is shown in the top-left corner, while the bar chart and tally occupy the central and bottom regions.

Additional Details

• Randomness: The coin flips are randomly generated but seeded for reproducibility in the animation.
• Smooth Transitions: All updates (bars, tally) use smooth transitions to avoid abrupt changes.
• Text Readability: The tally text has an opaque background to ensure readability against the bars.

Default Assumptions

• If no clarification is provided, the animation will use the following defaults:
  • 300 total flips (10 flips per second for 30 seconds).
  • Fair coin (50% probability for heads/tails).
  • Bar chart and tally as described.
  • No additional visual elements unless specified.