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Practice sheet · Video Games Path · Lens 1: Math

Vectors & chance

Practice for The Math of Video Games: vectors, frame rates, drop rates, expected value, DPS, pixels and gem prices (pp.8–16), plus Algebra A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

At least once: 1 − (chance of missing)ⁿ. A 2% drop misses 0.98 of the time. Expected value = add up chance × value. ms per frame = 1,000 ÷ fps.

Loot boxes are built around chance. The odds never “owe” you. Set a budget with an adult before you buy, and never chase losses.

Warm-up

One step

Vectors, frames and pixels.
1

Move it

A player at (200, 150) moves by (8, −6) in one frame. Where is it now, and how long is the vector?

2

Frame time

How many milliseconds per frame at 90 fps? At 30 fps?

3

Loot weights

A loot table has weights 120, 60, 18 and 2. What percent chance does each have?

4

Pixels

How many pixels are on a 2560 × 1440 screen? How many times as many as 1080p (2,073,600)?

Core

Two or three steps

Write the probability or formula first.
5

At least once

A rare drop has a 2% chance. What is the chance of getting it at least once in 50 tries? Round to a whole percent.

6

Fifty-fifty

For the same 2% drop, how many tries until you reach a 50% chance? Test n = 34 and n = 35.

Game Maths Practice · Video Games Path · Unizon1/2
Video Games Path · Unizon · PracticeVectors, chance & expected value
7

Is the box worth it?

A box costs 75 coins: 60% common (10), 30% rare (50), 9% epic (200), 1% legendary (2,000). Find its expected value. Who wins on average?

8

DPS

A weapon does 60 damage, 2 hits a second, with a 20% chance to crit for double. Find its average DPS. How long does a 1,800 HP boss last?

9

Armour

Use damage = attack × 100 ÷ (100 + armour) (p.12). A 150 attack hits armour 50, then armour 200. Find both.

10

Gem maths

A bundle is 1,200 gems for $11.99. What is the dollar price of a 900-gem item? Two items need how many bundles, and how many gems are left over?

Level Up

Equations

Lines and jump arcs (pp.17–18).
11

Runner

A runner starts at x = 60 and moves 250 px a second: x = 60 + 250t. When does it reach a door at x = 1,060?

12

The chase

An enemy starts at x = 760 and runs the same way at 150 px/s. When, and where, does the runner catch it?

13

Jump arc

A jump has height h = 10f − 0.25f² after f frames. Find the height at f = 20 and when the player lands.

14

Clear the ledge

When is the player 64 px up? Solve 10f − 0.25f² = 64. Why are there two answers?

Make your own problem

Your turn to be the game designer

  1. Invent a mechanic: a drop, a jump, a weapon, a shop.
  2. Write a question that uses a vector, a probability or an equation.
  3. Solve it, then ask: is it fair to the player?

My problem, and my worked answer:

Game Maths Practice · Video Games Path · Unizon2/2
Video Games Path · Unizon · PracticeVectors, chance & expected value

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Odds and prices are examples, not from a real game. Need to talk? Kids Help Phone: 1-800-668-6868.

Warm-up
1
(208, 144) · length 10 (200 + 8, 150 − 6). √(8² + 6²) = √100 = 10 px per frame.
2
≈ 11.1 ms · ≈ 33.3 ms 1,000 ÷ 90 ≈ 11.1. 1,000 ÷ 30 ≈ 33.3.
3
60% · 30% · 9% · 1% Total weight 200. 120 ÷ 200 = 60%, and so on.
4
3,686,400 · ≈ 1.78× 2,560 × 1,440 = 3,686,400. ÷ 2,073,600 ≈ 1.78.
Core
5
≈ 64% 0.98⁵⁰ ≈ 0.364 is the chance of missing all 50. 1 − 0.364 ≈ 0.636.
6
35 tries 1 − 0.98³⁴ ≈ 0.497 (just under) and 1 − 0.98³⁵ ≈ 0.507 (just over).
7
59 coins · the game 6 + 15 + 18 + 20 = 59. On average you lose 75 − 59 = 16 coins a box (example numbers).
8
144 DPS · 12.5 s 60 × 2 = 120. Crits add 20%: 120 × 1.2 = 144. 1,800 ÷ 144 = 12.5 s.
9
100 · 50 150 × 100 ÷ 150 = 100. 150 × 100 ÷ 300 = 50.
10
≈ $8.99 · 2 bundles · 600 left $11.99 × 900 ÷ 1,200 ≈ $8.99. 1,800 gems need 2 bundles ($23.98), leaving 600 to pull you back.
Level Up
11
4 s 60 + 250t = 1,060 → 250t = 1,000 → t = 4.
12
7 s at x = 1,810 60 + 250t = 760 + 150t → 100t = 700 → t = 7. x = 60 + 1,750 = 1,810.
13
100 px · lands at f = 40 h(20) = 200 − 100 = 100, the top. 10f − 0.25f² = 0 → f(10 − 0.25f) = 0 → f = 40.
14
f = 8 and f = 32 0.25f² − 10f + 64 = 0 → f² − 40f + 256 = 0 → f = (40 ± 24) ÷ 2. Once going up, once coming down.
Make your own: write the chance of missing first, then multiply; for money, turn gems back into dollars.
Game Maths Practice · Video Games Path · UnizonAnswer key