There are exactly three steps—Steps 1, 2, and 3, in that order—used to assemble light bulbs at a particular factory....

Phase 1: Owning the Dataset

Understanding the Flow

Let's create a flow diagram to understand how bulbs move through the assembly process:

Started → Step 1 → Step 2 → Step 3 → Finished
           ↓         ↓         ↓
         Failed   Failed   Failed

Key Definitions

• \(\mathrm{A}\) = bulbs making it through Step 1 ONLY (failed at Step 2)
• \(\mathrm{B}\) = bulbs making it through AT LEAST Step 2 (includes those that passed Step 2 but failed Step 3, AND those that passed all three steps)
• \(\mathrm{C}\) = bulbs making it through ALL THREE steps

Visual Representation

Let's use concrete numbers to test our understanding. Say we start with 100 bulbs:

Started: 100 bulbs
    ↓
After Step 1: Some pass, some fail
    - A bulbs pass Step 1 but fail Step 2
    - B bulbs pass at least Step 2
    ↓
After Step 2: Of the B bulbs
    - Some fail Step 3
    - C bulbs pass Step 3
    ↓
After Step 3: C bulbs completed

Phase 2: Understanding the Question

Breaking Down the Question

The manager wants: fraction of bulbs that made it through Step 1 that also made it through all three steps

Let's translate this:

• Numerator: bulbs that made it through all three steps
• Denominator: all bulbs that made it through Step 1

Key Insight

Which bulbs made it through Step 1?

1. Bulbs counted in \(\mathrm{A}\) (passed Step 1 only)
2. Bulbs counted in \(\mathrm{B}\) (passed at least Step 2, which means they must have passed Step 1)

Total bulbs through Step 1 = \(\mathrm{A + B}\)

Phase 3: Finding the Answer

Numerator

The numerator is straightforward: bulbs that made it through all three steps = \(\mathrm{C}\)

Denominator

All bulbs that made it through Step 1 = \(\mathrm{A + B}\)

• \(\mathrm{A}\) represents bulbs that passed Step 1 (but failed Step 2)
• \(\mathrm{B}\) represents bulbs that passed Step 1 AND Step 2 (some may have failed Step 3)
• Together, \(\mathrm{A + B}\) gives us all bulbs that passed Step 1

Phase 4: Solution

The fraction is: \(\frac{\mathrm{C}}{\mathrm{A+B}}\)

Numerator: \(\mathrm{C}\)
Denominator: \(\mathrm{A+B}\)