Visual Lessons

Predict one trial

Local seeded simulation of independent fair coin tosses. These are generated observations, not textbook experimental data. A fixed seed reproduces the same sequence.

Last 0 results: None yet

Long-run relative frequency tends towards the model value; exact equality at a set number of trials is not guaranteed.

Locate the chance

Even chance of purple versus not purple. Each card is equally selectable. The complementary event has 3 favourable cards.

Record, count, compare

Local seeded simulation of independent fair coin tosses. These are generated observations, not textbook experimental data. A fixed seed reproduces the same sequence.

Last 0 results: None yet

Long-run relative frequency tends towards the model value; exact equality at a set number of trials is not guaranteed.

Select an event

S = {1,2,3,4,5,6}. E = {2, 4, 6}. A check mark identifies each favourable outcome.

This counting rule applies because the six faces are equally likely. A biased die requires its outcome probabilities instead.

A longer run, not a guarantee

Local seeded simulation of independent fair coin tosses. These are generated observations, not textbook experimental data. A fixed seed reproduces the same sequence.

Last 0 results: None yet

Long-run relative frequency tends towards the model value; exact equality at a set number of trials is not guaranteed.

Who is included in the sample?

Illustrative population, not survey data: 100 people in equal groups A and B. In A, 40 of 50 prefer mango; in B, 10 of 50 do. The population share is 50%.

Build a complete sample space

n(S) = 4. Every possible recorded outcome appears once.

Uniform probabilities require fair independent coin tosses and, for cards, uniform independent selection.

Follow the conditional branches

Multiply conditional probabilities along a path. Add disjoint path probabilities for a larger event. At every possible node, the outgoing probabilities sum to 1.

Is the opposite result due?

Suppose the past six results were H H H H H H. For independent fair tosses, predict the next tail probability.

Fresh tosses: 0. Latest simulated result: not started. These fresh trials are not forced to offset the hypothetical history.

Change the replacement rule

n(S) = 12. 4 × 3 = 12; the four repeated-label pairs are excluded.

Each pair has probability $\frac{1}{12}$ under uniform selection at each draw.

Uniform landing and area

Assume uniform landing over the rectangle. Each local simulated point uses uniform horizontal and vertical coordinates. Boundary conventions do not change the continuous-area probability.

Area of circle ÷ area of rectangle: ($\frac{\pi }{4}$) ÷ 6. Without uniform landing, this ratio is not justified.