Every probability question in AP Statistics has the same shape once the words are stripped away: a distribution, a region, and a number. The tool below holds all three on screen at once, for the six distributions the course uses, with the TI-84 command that produces the same number printed underneath.

It is built for working, not for reading. Type the exact values from a problem into the parameter boxes, choose which region you want, and read the answer.

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Six distributions, typed parameters, four region shapes, and both directions: a region gives a probability, a probability gives a cutoff. The binomial keeps P(X = k) and P(X ≤ k) side by side, because those are the two the calculator distinguishes as binompdf and binomcdf and they are the pair most often swapped. The t curve carries the standard normal behind it in outline, so raising the degrees of freedom shows one collapsing onto the other. For the two sampling distributions the centre and spread are printed as expressions with the numbers substituted, and a warning appears when a normal-approximation condition fails.

What the two directions are called

The forward direction takes a region and returns a probability: normalcdf, tcdf, binomcdf. The inverse direction takes a probability and returns the boundary: invNorm, invT. Students conflate them constantly, and the tell is what the question hands you. If the problem gives you a score and asks what fraction of the population is below it, that is forward. If it gives you a percentage and asks which score sits at that percentile, that is inverse.

The calculator line under the readout always shows the command for whichever direction is selected, with the numbers already substituted, so the tool can be used to check a keystroke rather than only an answer.

The binomial keeps both forms on screen

binompdf gives \(P(X = k)\), the probability of exactly \(k\) successes. binomcdf gives \(P(X \leq k)\), the probability of at most \(k\). Both are always displayed for the current \(k\), because the swap between them is silent: the answer looks reasonable either way, and nothing in the arithmetic complains.

“At least 6” is the one worth rehearsing, since it needs both a complement and an off-by-one: \(P(X \geq 6) = 1 - P(X \leq 5)\), which is 1 - binomcdf(n, p, 5). Selecting “Right of a” with \(a = 6\) writes exactly that line.

Where the normal curve is standing in for something else

The two sampling-distribution settings draw a normal curve, but they are the only two that are drawing an approximation rather than the thing itself. The tool prints the centre and spread as expressions with the numbers substituted, and says so when a condition fails.

For a sample proportion the curve stands in for a binomial, and it is trustworthy when \(np \geq 10\) and \(n(1-p) \geq 10\). For a sample mean the curve is exact if the population is normal, and otherwise rests on the Central Limit Theorem and the \(n \geq 30\) guideline. Set \(n\) below either threshold and the warning appears; it is not decoration, it is the condition an exam response has to state.

A check worth performing before trusting any tool, including this one: set the normal to \(\mu = 0\), \(\sigma = 1\), choose “Between a and b” with \(a = -1.96\) and \(b = 1.96\), and confirm the readout says 0.9500. Then switch to t with \(df = 10\) and the same bounds; the area drops to 0.9216, because the t curve carries more of its probability in the tails. Raise \(df\) to 300 and it climbs back to 0.9491. That gap, and its closing, is the whole reason the course has two tables.