NUMB3RS - Season 3 - "Longshot" - Expected Value
Difficulty: Algebra 2 / Pre-Calculus / Statistics
Socratic Framework
In "Longshot," Danny Roberts utilized math to exploit racetrack betting's dynamic pari-mutuel systems. Let's study expected values in a controlled "geometric land-claim" system.
The Land Claim Framework
Imagine a class of 25 students deciding to co-own a large square plot of government-owned land. Each student invests a bid of c = \50,000$, resulting in a total combined investment pool of:
This entire square is divided into 8 distinct regions, labeled through . The total square area can be broken into 32 equal-sized grid units. The value of each region depends directly on its area (how many grid units it covers).
Because students do not know the boundaries beforehand, they place their marker randomly in the square. If a region has a larger area, it has a higher geometric probability of being chosen. If multiple students land in the same region, they must share its total value equally.
Geometric Land Claim Map
Total Area: 32 Grid Units. Click to drop claim markers!
Claim Statistics & expected Values
| Region | Grid Units | Probability (Decimal) | Regional Value | Simulated Claims |
|---|---|---|---|---|
| A | 4 | 0.1250 | $156,250 | 0 |
| B | 4 | 0.1250 | $156,250 | 0 |
| C | 2 | 0.0625 | $78,125 | 0 |
| D | 6 | 0.1875 | $234,375 | 0 |
| E | 4 | 0.1250 | $156,250 | 0 |
| F | 6 | 0.1875 | $234,375 | 0 |
| G | 3 | 0.0938 | $117,187.5 | 0 |
| H | 3 | 0.0938 | $117,187.5 | 0 |
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