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Solve estimation and probability brainteasers

ZooxData Scientist · Onsite
EasyData ScientistOnsiteStatistics & Math

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This question evaluates quantitative estimation, unit conversion, geometric scaling, and basic probability skills across multiple short brainteaser prompts.

This is a set of independent estimation and probability brainteasers of the kind asked rapid-fire from a slide deck. Treat each one separately. For every part, state any assumptions you make and show the back-of-the-envelope reasoning, not just the final number.

Clarifying Questions to Ask

For a rapid-fire brainteaser round like this, a few quick scoping questions show maturity before you dive in:

  • Should I show my reasoning out loud, or is only the final number expected? (For estimation problems the reasoning is usually the point.)
  • For the estimation parts, do you want an order-of-magnitude answer or a precise figure?
  • Am I allowed to state and use assumptions freely (e.g. movie length, frame rate, the Moon's angular size), or are specific inputs provided?
  • For the floor-tiling and storage parts, do you care about real-world layout/waste (cut tiles, overhead) or just the idealized arithmetic?
  • For the unit-heavy parts, which convention should I use — decimal vs. binary for "TB," and which definition of a "movie length"?

1) Tile a floor

A floor measures 15 ft × 20 ft. You cover it with square tiles, each of area 1 square yard.

Approximately how many tiles are needed?

2) Doubling account

An account starts with $1,000 and doubles every year.

How much is in the account after 20 years?

3) Estimate the Earth–Moon distance

You are sitting by a window looking at the Moon. Using reasonable "back-of-the-envelope" reasoning, estimate the distance from you to the Moon. State which inputs you assume (e.g., the Moon's angular size and physical diameter) and explain your method.

4) Stones in a boat

A boat is floating in a lake with some stones sitting in it. You throw the stones overboard into the lake.

Does the lake's water level rise, fall, or stay the same? Justify your answer.

5) Probability of an even product

There are two boxes. Each box contains three cards labeled 1, 2, 3 (one of each). You draw one card from each box, uniformly at random and independently.

What is the probability that the product of the two drawn numbers is even?

6) Uncompressed 1080p movies on a 2 TB drive

Approximately how many uncompressed 1080p movies fit on a 2 TB hard drive?

Clearly state your assumptions, including:

  • frame rate (fps),
  • color depth / bytes per pixel,
  • movie length, and
  • the definition of TB (decimal vs. binary).

7) Nested squares area

Three squares are nested concentrically, each inner square formed by connecting the midpoints of the sides of the square around it.

If the innermost square has area $A$, what is the area of the outermost square?

Constraints & Assumptions

  • The parts are mutually independent — solve each on its own; nothing carries over.
  • Estimation parts (1, 3, 6) are graded on stated assumptions and clean arithmetic, not a single "correct" number; an order-of-magnitude answer with sound reasoning is the goal. Reasonable defaults you may assume where not given: movie length ≈ 2 hours, frame rate 24–30 fps, 24-bit (3 bytes/pixel) color, and state whether "2 TB" means decimal ($2 \times 10^{12}$ B) or binary (TiB).
  • Exact parts (2, 4, 5, 7) have a single right answer — show the derivation, not just the figure.
  • Use consistent units throughout and call out conversions explicitly ($1\ \text{yd} = 3\ \text{ft}$; $1$ hour $= 3600$ s).

Hints

Hint 1

Units first: Both dimensions and the tile are easiest in the **same unit**. Recall $1\ \text{yd} = 3\ \text{ft}$, so a 1-square-yard tile is $3\ \text{ft} \times 3\ \text{ft}$. What's that in $\text{ft}^2$?

Hint 2

Area vs. layout: There are two defensible answers: a pure **area ratio** (floor area ÷ tile area, the material lower bound) and a **whole-tile layout** that rounds each dimension up to a whole number of tiles. Mention which one you mean — they differ here.

Hint 3

Doubling each year is geometric growth — after 20 years the multiplier is a power of two. Which power? The anchor $2^{10} = 1024 \approx 10^3$ lets you nail the order of magnitude fast.

Hint 4

What relationship to use: This is a **small-angle** problem. For a far object, angular size $\theta$ (in radians) relates physical diameter $D$ and distance $d$ by $\theta \approx D/d$. So which two inputs do you need to estimate?

Hint 5

The key anchor: The whole estimate hinges on the Moon's **angular size** — roughly **half a degree** (the Sun is about the same, which is why solar eclipses just barely fit). Convert that to radians before dividing. You'll also need a rough physical diameter for the Moon.

Hint 6

Compare displaced volumes: Apply **Archimedes** to the two situations separately — stones in the floating boat vs. stones resting on the lakebed — and ask what volume of water each displaces. Whether the level rises, falls, or holds depends on which displaced volume is larger. What governs that comparison?

Hint 7

Use the complement: A product is **even** unless it's odd, and a product of integers is odd **only when every factor is odd**. So compute $P(\text{both draws odd})$ and subtract from 1. (You can verify by enumerating all $3 \times 3$ ordered pairs.)

Hint 8

Build the byte budget: Multiply it out: $\text{pixels/frame} \times \text{bytes/pixel} \times \text{fps} \times \text{seconds}$ gives bytes per movie. Then divide the drive capacity by that. Resolution is $1920 \times 1080$; pick reasonable values for the other four assumptions.

Hint 9

Sanity-check the magnitude: Uncompressed 1080p is **enormous** — on the order of tens of GB per minute — so don't be surprised if only a small handful fit. State how the count shifts if you change fps, bit depth, or decimal-vs-binary TB.

Hint 10

The one fact you need: Connecting the **midpoints** of a square's sides produces a new square. What's the ratio of the new square's area to the original? (Find the new side via the right triangle with legs $s/2$, $s/2$.)

Hint 11

Count the steps: Each midpoint step multiplies area by a fixed factor. Going from outermost to innermost across **three** nested squares is **two** steps — so going *back* outward from area $A$ applies that factor's inverse twice.

Follow-up questions

  • After the main answers, expect the interviewer to probe how robustly you reason:
  • **Part 1:** How many tiles would you actually *order* once you account for cutting waste and breakage? How does the cut-tile count change if the tiles can't be cut at all?
  • **Part 3:** How sensitive is your Earth–Moon estimate to the angular-size assumption — if you were off by a factor of 2 on $\theta$, how far off is $d$? How would you measure the angular size more carefully with just your hand or a ruler?
  • **Part 5:** Generalize — if each box held cards $1$ through $n$, what's $P(\text{product even})$? What if you draw from $k$ boxes?
  • **Part 6:** Real codecs (H.264/H.265/AV1) compress video heavily. Roughly what compression ratio would make a typical 2-hour movie fit comfortably, and why is the uncompressed number so large?
  • **Part 7:** What's the area of the *n*-th nested square in terms of $A$? What does the side ratio (and rotation) converge toward?