Absolute and Relative Error in Rare-Event Approximations

Why Two Error Measures Matter

A Poisson approximation can be close to an exact binomial probability in absolute terms and still be far off in relative terms. This distinction matters especially for rare events: an error of a few ten-thousandths may look negligible on its own, but it may represent a large fraction of the probability being estimated.

For a binomial count \(X\sim\operatorname{Bin}(n,p)\), the standard Poisson approximation uses a Poisson random variable with parameter \(\lambda=np\). Let \(P\) denote the exact binomial probability of the event of interest and let \(Q\) denote its Poisson approximation. The two error measures compare \(Q\) with \(P\) in different ways.

Definition: The absolute error is \(|Q-P|\). When \(P>0\), the relative error, measured relative to the exact probability, is \(\dfrac{|Q-P|}{P}\). The relative error as a percentage is \(100\dfrac{|Q-P|}{P}\%\). Here the exact binomial probability is the benchmark.

Absolute error is in probability units. It tells you the size of the gap between the two probabilities, without regard to the size of the event probability. Relative error scales that gap by the exact probability, so it tells you how large the discrepancy is compared with the quantity you are trying to estimate. Relative error is undefined when \(P=0\); it should not be reported in that case.

Set Up the Comparison Correctly

For \(X\sim\operatorname{Bin}(n,p)\), the exact probability of an event \(A\) is calculated using the binomial distribution. For the Poisson approximation, use \(\lambda=np\) and calculate the same event under \(Z\sim\operatorname{Poisson}(\lambda)\). The event must be identical in both calculations: for example, \(X=5\) is compared with \(Z=5\), and \(X\geq 5\) is compared with \(Z\geq 5\).

P=P(X\in A),\qquad Q=P(Z\in A),\qquad Z\sim\operatorname{Poisson}(np)

The absolute error is symmetric: it is the same whether the approximation is above or below the exact probability. If the direction matters, also state whether \(Q>P\) (an overestimate) or \(Q<P\) (an underestimate). Relative error, as defined here, divides by \(P\), not \(Q\). Some other contexts use a different denominator, so a reported relative error should make its benchmark clear.

For a single count \(k\), the probabilities to compare are

P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}, \qquad P(Z=k)=e^{-np}\frac{(np)^k}{k!}.

For a tail such as \(X\geq r\), calculate the exact binomial tail and the Poisson tail over the same range, then compare those two event probabilities. Subtracting point probabilities at the threshold is not a valid substitute for comparing the tails.

Why Relative Error Can Be Large for a Rare Event

The relationship between the two measures follows directly from their definitions. If the exact probability \(P\) is positive, then

\text{relative error}=\frac{\text{absolute error}}{P}.

Thus, for a fixed absolute error, reducing \(P\) increases the relative error. This does not mean the approximation has become worse in every practical sense; it means that its discrepancy is large relative to the small probability being estimated. Which measure matters more depends on the question. If the task asks for probability accuracy in percentage points, absolute error may be directly useful. If it asks how accurate an estimate of a rare-event probability is proportionally, relative error is essential.

A short derivation makes the interpretation explicit. Write the approximation as \(Q=P+\Delta\), where \(\Delta\) is its signed error. Then \(|Q-P|=|\Delta|\), while dividing by the exact probability gives \(|\Delta|/P\). If, for example, \(P=0.0001\) and \(|\Delta|=0.0001\), the absolute error is only \(0.0001\), but the relative error is \(1\), or \(100\%\). The approximation's discrepancy is as large as the exact probability itself.

Worked Example: A Single Binomial Count

A hypothetical screening program independently tests \(100\) people, each of whom has probability \(p=0.01\) of producing a positive result. Let \(X\) be the number of positive results. Compare the exact and Poisson-approximated probabilities of exactly \(5\) positives, then calculate both errors. Use the binomial model as the exact benchmark.

Method. First calculate \(\lambda=np=100(0.01)=1\). Then evaluate the same event, \(X=5\), under the binomial and Poisson models.

Exact binomial probability.

P=P(X=5)=\binom{100}{5}(0.01)^5(0.99)^{95} =75{,}287{,}520(0.0000000001)(0.99)^{95} \approx 0.00289779.

The product of the first two factors is \(75{,}287{,}520(0.0000000001)=0.007528752\), and \((0.99)^{95}\approx0.3848964\). Thus \(P\approx0.007528752(0.3848964)=0.00289779\).

Poisson approximation. With \(\lambda=1\),

Q=P(Z=5)=e^{-1}\frac{1^5}{5!} =\frac{0.36787944}{120} \approx0.00306566.

Errors. The approximation is larger than the exact probability. The absolute error is \(0.00306566-0.00289779=0.00016787\). The relative error is \(0.00016787/0.00289779\approx0.05793\), or about \(5.793\%\).

Check. The signed difference is positive, as expected because \(Q>P\), and the relative error is less than \(100\%\) because the absolute error is smaller than \(P\). A common mistake is to divide by the Poisson estimate; that gives a different measure, not the relative error defined here.

Rare Tails: A Small Gap Can Be a Large Proportion

For a tail event, the exact and approximate probabilities may each be much smaller than one. In that setting, report enough digits to make the difference visible before rounding. Round only after calculating the errors; rounding both probabilities too early can distort a relative error.

Worked Example: A High-Threshold Tail

In a hypothetical batch of \(20\) independent units, each unit has probability \(p=0.05\) of having a specified defect. Let \(X\) count defective units. Compare the exact probability of at least \(5\) defects with the Poisson approximation, and calculate both errors.

Method. Use \(\lambda=np=20(0.05)=1\). For the exact binomial tail, it is convenient to calculate one minus the probabilities for \(0,1,2,3,4\) defects.

Exact binomial tail. The first probability is \(P(X=0)=0.95^{20}\approx0.35848592\). Use the successive-probability recurrence \(P(X=k+1)/P(X=k)=(20-k)(0.05)/((k+1)(0.95))\):

\begin{aligned} P(X=1)&=0.35848592\left(\frac{20(0.05)}{0.95}\right)\approx0.37735360,\\ P(X=2)&=0.37735360\left(\frac{19(0.05)}{2(0.95)}\right)\approx0.18867680,\\ P(X=3)&=0.18867680\left(\frac{18(0.05)}{3(0.95)}\right)\approx0.05958215,\\ P(X=4)&=0.05958215\left(\frac{17(0.05)}{4(0.95)}\right)\approx0.01332759. \end{aligned}

Therefore the probability of fewer than \(5\) defects is approximately \(0.35848592+0.37735360+0.18867680+0.05958215+0.01332759=0.99742606\), giving \(P=P(X\geq5)\approx1-0.99742606=0.00257394\).

Poisson approximation. For \(Z\sim\operatorname{Poisson}(1)\),

Q=P(Z\geq5) =1-e^{-1}\left(1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}\right) \approx 0.00365985.

The sum in parentheses is \(1+1+0.5+0.16666667+0.04166667=2.70833334\), so the Poisson tail is approximately \(1-(0.36787944)(2.70833334)=0.00365985\).

Errors. The absolute error is \(0.00365985-0.00257394=0.00108591\). The relative error is \(0.00108591/0.00257394\approx0.4219\), or about \(42.19\%\). The approximation overstates this tail.

Check. The absolute error is less than \(0.002\), but the relative error is over \(40\%\) because the exact tail is small. A common mistake is to call the approximation accurate based only on the small absolute difference. Whether this approximation is adequate depends on the required precision and the use of the probability.

Very Small Probabilities and Percentage Errors

A probability can be small enough that a modest absolute discrepancy produces a striking percentage error. This does not change the definitions: calculate the absolute difference first, then divide by the exact probability. State the scale clearly, especially when reporting several leading zeroes.

Worked Example: A Rare Exact Count

A hypothetical monitoring system records \(10\) independent opportunities for a rare alert, with probability \(p=0.01\) of an alert on each opportunity. Let \(X\) be the alert count. Compare the probabilities of exactly \(3\) alerts under the exact binomial model and its Poisson approximation, and calculate both errors.

Method. Here \(n=10\), \(p=0.01\), and \(\lambda=np=0.1\). Calculate the exact point probability and the Poisson point probability at \(3\).

Exact probability.

P=P(X=3)=\binom{10}{3}(0.01)^3(0.99)^7 =120(0.000001)(0.93206535) \approx0.00011185.

Poisson approximation.

Q=e^{-0.1}\frac{(0.1)^3}{3!} =(0.90483742)\frac{0.001}{6} \approx0.00015081.

Errors. The absolute error is \(0.00015081-0.00011185\approx0.00003896\). The relative error is \(0.00003896/0.00011185\approx0.3483\), or about \(34.83\%\). The Poisson approximation overstates the exact probability.

Check. The absolute error is less than \(0.00004\), while the relative error is nearly \(35\%\). A common mistake is to report only the absolute error and conclude that the difference is unimportant, without considering how small the target probability is.

Practical Interpretation and Exam Traps

Both measures are useful, but they answer different questions. Absolute error directly measures the probability gap. Relative error measures that gap against the exact probability and can be especially informative for rare events. When presenting an approximation, a clear comparison usually includes the exact probability, the approximate probability, the absolute error, and the relative error as a percentage.

Question	Useful measure	Interpretation
How far apart are the probabilities?	Absolute error	The difference in probability units
How large is the error compared with the exact event probability?	Relative error	The absolute error divided by the exact probability
Does the approximation overstate or understate the event probability?	Compare \(Q\) with \(P\)	\(Q>P\) is an overestimate; \(Q<P\) is an underestimate

A frequent trap is using the Poisson approximation as the denominator in the relative error when the requested benchmark is the exact binomial probability. Another is mixing events: an exact probability for \(X\geq r\) cannot be compared with an approximate probability for \(Z=r\). A third is rounding both probabilities to too few decimal places before subtracting, particularly when the probabilities are small.

Key takeaway: Use the exact binomial probability as the benchmark: absolute error is \(|Q-P|\), and relative error is \(|Q-P|/P\) when \(P>0\). For rare events, always consider both; a small absolute error can be a large percentage of the exact probability.

Check Your Understanding

For each problem, treat the exact binomial probability as the benchmark unless stated otherwise. Round probability and error values to six decimal places and relative errors to four decimal places when needed.

A count has distribution \(X\sim\operatorname{Bin}(10,0.1)\). Approximate \(P(X=0)\) with a Poisson model. Find the absolute and relative errors. (Answer: Exact \(0.348678\), approximation \(0.367879\), absolute error \(0.019201\), relative error \(0.0551\), or \(5.51\%\).)
Let \(X\sim\operatorname{Bin}(4,0.1)\). For the event \(X=4\), calculate the exact probability and the Poisson approximation, then find both errors. (Answer: Exact \(0.000100\), approximation \(0.000715\), absolute error \(0.000615\), relative error \(6.1501\), or \(615.01\%\).)
An exact event probability is \(0.000080\), and its Poisson approximation is \(0.000120\). Find both error measures and state whether the approximation overstates or understates the probability. (Answer: Absolute error \(0.000040\); relative error \(0.5000\), or \(50\%\); it overstates the probability.)
For \(X\sim\operatorname{Bin}(50,0.02)\), compare the exact probability \(P(X=0)\) with its Poisson approximation. Find the absolute and relative errors. (Answer: Exact \(0.364170\), approximation \(0.367879\), absolute error \(0.003710\), relative error \(0.0102\), or about \(1.02\%\).)
An exact event probability is \(0.000200\) and an approximation is \(0.000250\). Calculate relative error first using the exact probability as denominator, then using the approximation as denominator. Explain why the answers differ. (Answer: Exact-denominator relative error \(0.25\), or \(25\%\); approximation-denominator measure \(0.20\), or \(20\%\). The denominators differ.)

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