Roll dice thousands of times and watch the messy experimental frequencies settle onto the exact theoretical answer. Probability is what happens in the long run — not what happens next.
The chart compares the experimental frequency with the theoretical probability as trials accumulate.
Counting outcomes, combining events, and why a fair coin owes you nothing.
Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). When every outcome is equally likely it is pure counting:
The sample space is the complete list of things that could happen — the tiles shown under the chart. An event is any subset of it: "even number", "greater than 4". Selecting events A and B highlights their tiles, and tiles in both are the intersection.
Getting the sample space right is most of the work. For two dice there are 36 equally likely outcomes but only 11 possible sums — and those sums are not equally likely, which is why the two-dice chart is a triangle rather than a flat line.
That subtraction is the part people forget. Adding P(A) and P(B) double-counts every outcome living in both events, so you remove one copy. Pick two overlapping events and check the arithmetic against the live numbers.
When events cannot happen together — mutually exclusive — the overlap is 0 and the rule simplifies to plain addition.
Two events are independent when knowing one happened tells you nothing about the other. The test is arithmetic:
The app checks this for your chosen events and reports the verdict. It is a genuinely surprising result on a single die: "even" and "greater than 3" are not independent, while "even" and "greater than 4" happen to be — independence is a property of the numbers, not of intuition.
Press Run continuously. Early on the experimental line jumps around wildly; after a few thousand trials it settles onto the theoretical value and stays there.
That is the law of large numbers: averages converge in the long run. What it emphatically does not say is that outcomes "balance out". After ten heads, the next flip is still exactly 50/50 — the coin has no memory. Convergence happens because early flukes are diluted by the growing pile of trials, not because anything corrects them.
Count the outcomes, define events as subsets, subtract the overlap when combining with OR, and multiply only when events are genuinely independent. Probability describes long-run behaviour — it makes no promise whatsoever about your next roll.
Roll dice thousands of times and watch the messy experimental frequencies settle onto the exact theoretical answer. Probability is what happens in the long run — not what happens next.