Approximation error and decision margins

Construct exact finite draw cases, measure a replacement approximation’s error and separate numerical precision from decision agreement.

19 minutes · advanced · Original Poker4Quant lesson

What you will learn

Audit an approximation in percentage points, test a stated precision tolerance and distinguish actual threshold agreement from a sufficient interval certificate.

Prerequisites: Replacement and sampling protocols · Monte Carlo estimates · Decision robustness

Name the quantity and approximation separately

Suppose two cards are sampled uniformly without replacement from a toy deck of N distinct cards. K cards are marked targets and U=N−K are unmarked. We want the probability of at least one target. This event is not a poker win probability: neither ranks nor opponent hands determine the target flag. The exact probability follows by excluding two unmarked draws.

There are N(N−1) equally likely ordered physical pairs. Exactly U(U−1) miss twice, so p=1−U(U−1)/[N(N−1)]. Reusing the first-draw miss rate gives q=1−(U/N)². That replacement expression is a deliberate approximation to the removal model. It is exact for a different sampling protocol, not an algebraic rearrangement of the same one. Manual case inspection constructs an exact set; it is not a random trial average.

Give error a unit and an explicit tolerance

Absolute probability error is |p−q|. Expressing both probabilities as percentages makes their difference an error in percentage points. For p=50% and q=43.75%, the absolute error is6.25percentage points. Relative error would instead divide the probability error by the nonzero exact value:0.0625/0.5=12.5%. These are different reports; neither should be silently substituted for the other.

An approximation meets an absolute precision requirement only when its error is at most the stated tolerance. The tolerance must have the same units. In our finite examples exact counting supplies the reference, and p−q=K(N−K)/[N²(N−1)]. This is deterministic model mismatch, not floating-point roundoff or Monte Carlo sampling noise. Do not use extra simulation trials to claim the wrong replacement model has become exact.

A threshold can turn a numerical gap into a different action

Use a supplied educational rule: act when target probability is at least a cutoff c; otherwise decline. The rule includes equality. It is not a recommendation to bet, trade or spend money. Under N=4,K=1, exact p=50% while approximate q=43.75%. At cutoff48%, exact counting says act and the approximation says decline. A probability report that looks close can change a near-boundary action.

At N=5,K=1, p=40% and q=36%. Both decline at cutoff50%, although the4percentage-point error fails a2point precision requirement. Correct action agreement does not validate the numerical estimate. Conversely, meeting an error tolerance by itself does not establish agreement for every possible cutoff. Ask which claim the application actually needs.

A sufficient interval certificate is not necessary agreement

If a valid absolute bound δ guarantees |p−q|≤δ, the symmetric interval [q−δ,q+δ], clipped to[0,1], contains p. It certifies acting under the inclusive rule when its lower endpoint is at least c. It certifies declining only when its upper endpoint is strictly below c. Touching the cutoff from below is insufficient: equality would require acting. An interval crossing the cutoff gives no certificate, even if the actual exact decision happens to agree.

At N=6,K=3, exact p=80%, q=75% and error5percentage points. The symmetric interval[70%,80%]certifies acting at cutoff70%, including its lower boundary. It also meets the supplied6point precision requirement. In these teaching models δ is known from exact counting. A real application would need a valid independently justified bound; guessing a margin is not a certificate. The symmetric test deliberately discards the known sign p≥q in this particular draw approximation, so it is conservative rather than the strongest possible directional test. It is not a statistical confidence interval.

Worked example

A five-card toy deck has one target. Compare exact removal with the replacement approximation under cutoff50% and precision tolerance2percentage points.

The four unmarked cards create4×3=12ordered double misses among5×4=20ordered pairs. Exact hit probability is1−12/20=40%.
Replacement uses(4/5)²for a double miss, giving q=1−16/25=36%. The absolute gap is4percentage points.
Both40%and36%fall below50%, so their actual decisions agree: decline.
The error4exceeds the2point tolerance. A symmetric4point interval around36%is[32%,40%], wholly below50%, so it certifies declining even though the stricter precision requirement fails.

Keep probability accuracy, actual action agreement and a sufficient error-bound certificate as separate claims with explicit units.

Hands-on: Construct exact draw cases and audit an approximation

Inspect distinct physical-card pairs, build every double miss, audit absolute percentage-point error and separate precision tolerance, exact/approximate actions and a sufficient interval certificate.

Reading approximation workshop…

Common mistakes

Calling percentage-point error relative percent error.
Using a replacement model without naming the changed sampling law.
Treating deliberately selected case inspections as a random sample.
Counting the same physical card twice in a without-replacement pair.
Inferring a precise probability because the decision happens to agree.
Claiming that failure of a sufficient interval test proves a wrong action.
Treating an error bound as a confidence interval or a guessed margin as a guarantee.

Keep these ideas

Construct the exact event and state the approximating law.
Report absolute error and its units before checking tolerance.
Apply the decision cutoff to each probability separately.
Use a valid interval bound and the rule’s equality convention for a sufficient certificate.