Exact Probabilities and Normal Approximations
In Normal Approximation to the Binomial, you learned how to use a normal curve to estimate binomial probabilities when the Large Counts condition is satisfied. An estimate can be convenient, especially for cumulative probabilities, but it is not the same as the exact probability from the binomial model. Comparing the two helps you see how close the approximation is for a particular event.
This tutorial compares exact and approximate probabilities for a binomial count with \(n=50\). We will use one model and examine three kinds of events: one exact count, a cumulative event, and an inclusive range. The same model can produce approximation errors of different sizes for different events.
For a binomial setting, the exact probability of exactly \(k\) successes can be found with the binomial formula or \(\operatorname{binompdf}(n,p,k)\). Cumulative and range probabilities can be found with \(\operatorname{binomcdf}\), as in the earlier tutorials on binomial probabilities. For a normal estimate, use the continuity correction from the previous tutorial.
How to Compare the Results
A useful comparison reports both values and their difference. The absolute error is the nonnegative distance between the approximate probability and the exact probability. It tells us how far apart the two answers are in probability units. It does not tell us whether the approximation is an overestimate or an underestimate, so also state which value is larger.
Keep full calculator precision during the calculations, then round the reported probabilities and error consistently. If you subtract already-rounded probabilities, your error may differ in the last digit from the error calculated using the unrounded values.
The Large Counts condition supports using the normal model; it does not guarantee a particular level of accuracy for every event. Accuracy depends partly on the event being estimated. A point probability, for example, comes from a narrow interval under the normal curve, while a cumulative probability combines many possible binomial counts.
Worked Example: Exactly 25 Successes
Worked Example: Exactly 25 Successes
Suppose a hypothetical simulation runs 50 independent trials, each with success probability 0.5. Let \(X\) count the number of successes. Compare the exact probability of exactly 25 successes with its normal approximation.
State. The model is \(X\sim B(50,0.5)\), and the event is \(X=25\).
Plan. Each trial has two outcomes, the number of trials is fixed at 50, the trials are independent, and the success probability is constant at 0.5. These are the BINS conditions, so the binomial model is appropriate under the stated assumptions. For the normal approximation, \(np=50(0.5)=25\) and \(n(1-p)=50(0.5)=25\). Both are at least 10, so the Large Counts condition is satisfied.
Do. First calculate the exact probability with the binomial formula:
The calculator check is \(\operatorname{binompdf}(50,0.5,25)\approx0.1122752\). The model’s mean and standard deviation are:
To estimate \(P(X=25)\) with a normal curve, the continuity-corrected event is \(24.5<Y<25.5\), where \(Y\) has mean 25 and standard deviation \(\sqrt{12.5}\). Thus:
The normal estimate is slightly larger than the exact probability. Using the unrounded calculator values, the absolute error is:
More precisely, the difference from the unrounded probabilities is approximately 0.0001877433, which rounds to 0.0001877. As a check on the normal calculation, the corrected boundaries are each \(0.5/\sqrt{12.5}\approx0.1414\) standard deviations from the mean, giving the same area as the standard normal area between \(-0.1414\) and \(0.1414\).
Conclude. Under this model, the exact probability of 25 successes is about 0.1123, and the continuity-corrected normal estimate is about 0.1125. The normal approximation overestimates the exact probability by approximately 0.0001877.
Worked Example: At Most 25 Successes
Worked Example: At Most 25 Successes
Use the same hypothetical simulation: 50 independent trials, each with success probability 0.5, and \(X\) counts the successes. Compare the exact probability of at most 25 successes with the normal approximation.
State. Here \(X\sim B(50,0.5)\), and the event is \(X\leq25\).
Plan. The BINS conditions hold under the stated model: there are two outcomes per trial, the fixed number of trials is 50, the trials are independent, and \(p=0.5\) stays the same. Also, \(np=25\) and \(n(1-p)=25\), so the Large Counts condition is satisfied.
Do. Find the exact cumulative probability with \(\operatorname{binomcdf}\):
This exact value can also be checked using symmetry. For \(B(50,0.5)\), the probabilities on either side of 25 match in pairs. Therefore, the probability through 25 equals half of 1 plus the probability at 25:
For the normal approximation, \(Y\) has mean 25 and standard deviation \(\sqrt{12.5}\). Since “at most 25” includes 25, the continuity-corrected boundary is 25.5:
The normal estimate is again a little larger. Subtracting the unrounded values gives an absolute error of approximately 0.0000939. As another check, the normal probability is the area to the left of a boundary \(0.5/\sqrt{12.5}\approx0.1414\) standard deviations above the mean.
Conclude. The exact probability of at most 25 successes is about 0.5561, while the normal approximation is about 0.5562. In this case, the approximation overestimates the exact probability by approximately 0.0000939.
Worked Example: From 24 Through 26 Successes
Worked Example: From 24 Through 26 Successes
For the same 50-trial model, compare the exact probability of getting from 24 through 26 successes, inclusive, with the normal approximation.
State. The random variable is \(X\sim B(50,0.5)\), and the event is \(24\leq X\leq26\).
Plan. The binomial model’s BINS conditions hold as described above. The Large Counts values are \(np=25\) and \(n(1-p)=25\); each is at least 10. We can calculate the exact range probability by adding the three point probabilities, then compare it with the continuity-corrected normal area.
Do. The exact probability is the sum for 24, 25, and 26 successes. Symmetry gives \(P(X=24)=P(X=26)\), and the binomial formula gives \(P(X=24)=P(X=25)(25/26)\):
Therefore, adding the three exact point probabilities gives:
This can be checked directly with \(\operatorname{binomcdf}(50,0.5,26)-\operatorname{binomcdf}(50,0.5,23)\), which gives approximately 0.3281890. The subtraction uses 23 as the lower cumulative cutoff so that the included counts begin at 24.
The inclusive range gets a continuity correction at both ends: \(23.5<Y<26.5\). The mean is 25 and the standard deviation is \(\sqrt{12.5}\), so:
The normal estimate is larger than the exact probability. The absolute error from the unrounded values is approximately 0.0004378. This error is larger than the errors in the first two examples, even though the same binomial model is used.
Conclude. The exact probability of 24, 25, or 26 successes is about 0.3282. The normal approximation is about 0.3286, an overestimate by approximately 0.0004378.
What the Comparisons Tell Us
All three normal estimates are close to their exact binomial probabilities, but they are not identical. In these examples, each normal estimate is larger than the exact value. That direction is a feature of these particular comparisons, not a rule that every normal approximation must overestimate. A different event or a different value of \(p\) may produce a smaller estimate instead.
| Event | Exact binomial probability | Normal approximation | Absolute error |
|---|---|---|---|
| \(X=25\) | 0.1122752 | 0.1124629 | 0.0001877 |
| \(X\leq25\) | 0.5561376 | 0.5562315 | 0.0000939 |
| \(24\leq X\leq26\) | 0.3281890 | 0.3286268 | 0.0004378 |
The absolute errors are all small compared with the probabilities in this table. But “small” should be judged in relation to the purpose of the calculation. If a decision depends on distinguishing probabilities that differ by less than 0.0005, an error of about 0.0004 could matter. When a calculator can easily provide the exact binomial probability, use it when exactness is requested or when the difference could affect the conclusion.
The continuity correction matters because it helps the continuous normal curve cover the whole-number counts in the event. It improves the connection between a discrete binomial event and a continuous area, but it does not force the two distributions to match perfectly. The normal curve has a different shape from the binomial probability distribution, so some discrepancy can remain even when the Large Counts condition is satisfied.
Common Mistakes and AP Exam Tips
- Calling an approximation exact. Identify the normal result as an estimate or approximation. The exact value comes from the binomial distribution or a binomial calculator command.
- Using the wrong event boundary. For \(X\leq25\), the continuity-corrected boundary is 25.5 because the event includes 25. For \(24\leq X\leq26\), use 23.5 and 26.5.
- Comparing rounded values too early. Keep the calculator’s full precision for the subtraction. Round the probabilities and the absolute error only when reporting the final results.
- Reporting only the absolute error. The absolute error has no direction. State whether the normal estimate is larger or smaller than the exact probability.
- Assuming the same error for every event. The point, cumulative, and range probabilities above have different absolute errors. Evaluate the event being asked about rather than making a general accuracy claim from one comparison.
- Forgetting what the probability describes. State the event in context—for example, the probability of exactly 25 successes in 50 trials—not merely “the answer is 0.1125.”
For full-credit communication, define \(X\), identify \(n\) and \(p\), verify the relevant binomial and Large Counts conditions, and show both methods. For the normal estimate, explicitly write the continuity-corrected event. Then report the exact probability, the approximate probability, and the difference in context.
Key Takeaway
The normal approximation can be close to the exact binomial probability without being exact. In a model with 50 trials and success probability 0.5, comparing point, cumulative, and range events shows why accuracy should be checked for the event at hand.
Check Your Understanding
Use \(X\sim B(50,0.5)\) for the questions below. When asked to compare probabilities, report the direction of the difference as well as its size.
- For \(P(X=25)\), write the continuity-corrected normal event and identify the exact calculator command.
- For \(P(X\leq20)\), what upper boundary should be used for the continuity-corrected normal approximation?
- Write a binomcdf expression that calculates \(P(22\leq X\leq28)\).
- Why does satisfying the Large Counts condition not guarantee that an approximate probability equals the exact binomial probability?
- If an exact probability is 0.20 and its normal estimate is 0.198, find the absolute error and state whether the estimate is an overestimate or an underestimate.