Half full

A half is a third of it. What is it?

2026 Update: The Half-Full Glass — Estimation and Fermi Problems

The classic “is the glass half full or half empty?” can be modeled mathematically and connects to Fermi estimation — a key skill in system design interviews at Google, Jane Street, and quant firms.

def glass_measurement(fill_fraction: float, optimistic: bool = True) -> str:
    """
    Model the half-full/half-empty perspective mathematically.
    A glass is 'half full' from an optimistic perspective (gaining),
    'half empty' from a pessimistic perspective (losing).
    """
    if fill_fraction > 0.5:
        return f"More than half: {fill_fraction:.1%} full"
    elif fill_fraction  float:
    """
    Estimate molecules in volume_ml of water.
    Water density: 1 g/ml
    Molar mass of H2O: 18 g/mol
    Avogadro's number: 6.022e23 molecules/mol
    """
    water_mass_g = volume_ml  # Since density = 1 g/ml
    moles = water_mass_g / 18
    avogadro = 6.022e23
    molecules = moles * avogadro
    return molecules

molecules = water_molecules_in_glass(125)  # Half a 250ml glass
print(f"Molecules in half-full glass: {molecules:.3e}")  # ~4.18e24

# Fermi estimation framework for tech interviews
def estimate_daily_youtube_uploads() -> dict:
    """
    Fermi estimate: How many hours of video are uploaded to YouTube daily?
    Fact-check: YouTube says 500 hours/minute = 720,000 hours/day.
    """
    global_internet_users = 5e9
    youtube_users_fraction = 0.5   # ~50% of internet users use YouTube
    daily_active_users = global_internet_users * youtube_users_fraction * 0.3  # 30% daily active
    fraction_who_upload = 0.001    # 0.1% upload content
    avg_upload_per_uploader_minutes = 5
    hours_per_day = (daily_active_users * fraction_who_upload *
                     avg_upload_per_uploader_minutes) / 60
    return {
        "estimated_hours_per_day": hours_per_day,
        "actual_hours_per_day": 720_000,
        "order_of_magnitude_error": abs(
            round(hours_per_day / 720_000 if hours_per_day > 720_000
                  else 720_000 / hours_per_day)
        )
    }

result = estimate_daily_youtube_uploads()
for k, v in result.items():
    print(f"  {k}: {v:,.0f}")

Fermi estimation in tech interviews:

• “How many requests per second does Google Search handle?” (~8.5B queries/day ≈ 100K/sec)
• “How much storage does Gmail need for 2 billion users?” (avg email ~75KB × 50 emails/day × 365 × 2B users)
• “How many servers does Netflix need?” (decompose by: users, concurrent streams, bitrate, cache hit rate)

The skill is structured decomposition: identify the key variables, make explicit estimates for each with sanity checks, and combine. An answer within 1-2 orders of magnitude is considered excellent.