Probability Analysis: Horse Race Mathematics

[hasjack]

[horse82]

Race Probability Analysis

I analyzed the probability of not winning the horse race based on the code mechanics. Here's what I found:

Race Mechanics

• Player horse speed: random(2,9) pixels/tick
• AI horse speed: random(2,8) pixels/tick
• Movement interval: 50ms
• Race distance: 1410 pixels (from x=580 to x=1990)

Mathematical Analysis

Using the Central Limit Theorem and probability theory:

1. Per-frame statistics:

   • Player: mean = 5.5, variance = (7^2)/12 ≈ 4.08
   • AI: mean = 5.0, variance = (6^2)/12 ≈ 3.00
   • Combined std dev per frame = √(4.08 + 3.00) ≈ 2.65

2. Probability calculations:
   For n frames, Z-score = (n * 0.5) / (√n * 2.65)
   P(losing) = P(Z < -z_score)

Results

Probability of not winning:

• 10 frames: 27.5% (Z = -0.597)
• 100 frames: 2.94% (Z = -1.89)
• 256 frames: 0.122% (Z = -3.03)
• 1000 frames: 0.000000118% (Z = -5.97)

Game Design Implications

This creates perfect game feel:

1. Early race is competitive (27.5% behind chance)
2. Mid-race shows your advantage (2.94%)
3. Victory feels earned but nearly certain (0.122%)

The exponential decay in probability makes the race exciting while ensuring player satisfaction - a masterclass in game design through mathematics.