Phase 1: Owning the Dataset
Understanding the System
Let's map out the vending machine's token exchange system:
Token Values:
- \(1 \text{ large token} = 5 \text{ small tokens}\)
- \(\text{Beverage price} = 3 \text{ small tokens}\)
Payment Options (equally likely):
- Pay with 3 small tokens → Get beverage
- Pay with 1 large token → Get beverage + 2 small tokens change
Initial State (after last service):
- Small tokens in machine: 50
- Large tokens in machine: 0
- Beverages sold since then: 400
Visual Representation
Let's create a flow diagram showing token movements:
VENDING MACHINE XJ100
┌─────────────────────────────────┐
│ Start: 50 small, 0 large │
│ │
│ For each beverage: │
│ ┌─────────────┬───────────────┐ │
│ │ 50% chance │ 50% chance │ │
│ │ │ │ │
│ │ IN: 3 small │ IN: 1 large │ │
│ │ OUT: 0 │ OUT: 2 small │ │
│ │ │ │ │
│ │ Net: +3S │ Net: +1L, -2S │ │
│ └─────────────┴───────────────┘ │
│ │
│ 400 beverages sold │
└─────────────────────────────────┘
Phase 2: Understanding the Question
We need to find the expected number of tokens of each type in the machine today.
Key Insight
Since customers are equally likely to use either payment method, we can calculate the expected change in tokens per beverage, then multiply by 400.
Expected change per beverage:
- Small tokens: \(0.5 \times (+3) + 0.5 \times (-2) = 1.5 - 1 = +0.5\)
- Large tokens: \(0.5 \times (0) + 0.5 \times (+1) = 0 + 0.5 = +0.5\)
Phase 3: Finding the Answer
Systematic Calculation
Starting from the initial state and adding the expected changes:
Small tokens:
- Initial: 50
- Change from 400 beverages: \(400 \times 0.5 = 200\)
- Total expected: \(50 + 200 = 250\)
Large tokens:
- Initial: 0
- Change from 400 beverages: \(400 \times 0.5 = 200\)
- Total expected: \(0 + 200 = 200\)
Verification
Let's verify our logic:
- 200 beverages (expected) paid with 3 small tokens: \(200 \times 3 = 600\) small tokens in
- 200 beverages (expected) paid with 1 large token: 200 large tokens in, 400 small tokens out as change
- Net small tokens: \(50 + 600 - 400 = 250\) ✓
- Net large tokens: \(0 + 200 = 200\) ✓
Phase 4: Solution
Based on our analysis:
- Small tokens: 250
- Large tokens: 200
These values reflect the expected token counts after 400 beverages were sold with equal probability of each payment method.
