Making Informed Decisions — Visual Lab

What you're seeing

The grid on the left is the sample space — every equally-likely outcome of the experiment you picked. Each cell is shaded by which events it belongs to: solid means it's in both A and B, a medium shade means A only, a faint shade means B only, and white means neither.

The two-way table just counts those shaded cells, with row and column totals. Every probability on the right is a count divided by the grand total — including the conditional \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\), which is the same as dividing the "A and B" count by the "B" total. The independence flag checks whether \(P(A\cap B)=P(A)\,P(B)\). At the bottom, the convergence chart plots your running experimental \(P(A)\) against the dashed theoretical line.

Try this

1. Start on One fair die. Find the cells that are in both "even" and "greater than 3" — confirm the table shows \(P(A\text{ and }B)=\tfrac{2}{6}\).
2. Compare \(P(A)\) with \(P(A\mid B)\). When they differ, knowing B changed the odds — the events are not independent. Check the flag.
3. Switch to Two fair coins and confirm the flag reads Independent: \(P(A\text{ and }B)=\tfrac14=\tfrac12\cdot\tfrac12\).
4. Press +25, then +500 a few times. Watch the experimental bar and the curve drift toward the theoretical line — few trials are noisy, many trials settle.