Tools & Resources
Curated extras for Topic 6: interactive tools, supplemental videos, and references. Pick what helps. Skip what doesn't.
Hands-on widgets for poking at the math. Drag a slider, watch the line move.
Expected Value Lab
Define each outcome of a chance situation — value and probability — and watch E(X) compute live. Try the carnival, lottery, and insurance presets to see why every gambling operation is engineered for a negative expected value to the player.
Law of Large Numbers Simulator
Pick coin, die, or a custom probability, then run trials in batches of 10, 100, 1,000, or 10,000. Watch the experimental probability bounce around at low n and converge on the theoretical value as n grows — the Law of Large Numbers in real time.
Probability explained — Independent and dependent events
Sal opens with the coin and die — favorable outcomes over total outcomes — and establishes the 0-to-1 range. The cleanest intro to P(event) for a first-time learner. Evergreen despite being older.
Probability with playing cards
Direct application of the favorable/total ratio to a standard 52-card deck — sets up the deck framework students will see again in Lesson 6.3.
The Fundamental Counting Principle
Multiple worked examples (clothing combinations, license plates) walking through the multiplication-across-stages logic. Tight, exam-ready treatment.
Fundamental Counting Principle, Tree Diagrams, and Probability
Builds tree diagrams alongside the multiplication rule — the visual companion that Khan and OCT don't lean into as heavily. A short reach outside the preferred list to cover the tree-diagram bonus.
Addition rule for probability
Sal derives P(A or B) = P(A) + P(B) − P(A and B) using Venn diagrams and explains why we subtract the overlap. Crisp foundational treatment of the formula.
Probability with playing cards and Venn diagrams
The exact 'P(red card or king)' worked example, with both the addition rule and a Venn diagram showing the overlap subtraction step explicitly.
Expected profit from lottery ticket
Sal applies E(X) = Σ value × probability to a lottery ticket, computes the expected profit, and shows why the EV is negative for the player (positive for the lottery). Hits the formula and the house-edge insight in one shot.
Expected payoff example — lottery ticket
2021 Khan reshoot of the same concept with cleaner audio. Functionally interchangeable with the above; pick whichever production quality fits.
Experimental versus theoretical probability simulation
Sal contrasts theoretical and experimental probability and names the Law of Large Numbers as the bridge: as trials increase, experimental probability converges to theoretical.
Law of large numbers
Deeper treatment of LLN with coin flips as the running example. Older Sal video but the convergence argument is evergreen — pairs well with the simulation video above.