Coincidence or Anomaly? Understanding Probability in Unexplained Events

A team is investigating an old building. At 10:17 p.m., a light flickers. A few seconds later, someone hears a short knock. At almost the same moment, an investigator notices a sudden change on a handheld meter.

The team looks at one another. Three unusual things happened together. Surely that combination must mean something.

But there is another possibility: the events may have occurred by chance, may have had ordinary causes, or may only appear connected because the team noticed them together.

This is where probability becomes important.

Probability does not make an investigation less interesting. It helps us ask a more precise question: how surprising would this observation be if ordinary processes were operating?

A coincidence can be unusual without being impossible. A rare event is not automatically evidence of a paranormal cause. To evaluate an unexplained event responsibly, investigators need to consider how often similar events occur, how many opportunities there were for them to happen, what was measured, and whether the observation was defined before it occurred.

This article introduces practical statistical ideas that can help AERM assess unusual reports without exaggerating what the numbers show.

1. What Is Probability?

Probability is a mathematical way of describing uncertainty about an event.

It is commonly expressed as a number between 0 and 1:

• 0 means an event is impossible under the model being used.
• 1 means an event is certain under the model.
• A value between 0 and 1 represents a degree of likelihood.

Probability can also be expressed as a percentage. For example, a probability of 0.25 is equivalent to 25%.

For a simple experiment with equally likely outcomes:

\[ P(A) = \frac{\text{Number of outcomes favourable to A}}{\text{Total number of possible outcomes}} \]

Consider a fair six-sided die. The probability of rolling a 4 is:

\[ P(4) = \frac{1}{6} \approx 0.167 \]

That is approximately 16.7%.

A single roll producing a 4 is not surprising. Even if a result has a relatively low probability, it can still occur during ordinary random variation.

Probability is not an explanation

Probability describes uncertainty within a defined model. It does not automatically tell us why an event occurred.

If a light flickers during an investigation, assigning a probability to that event requires a model of how often the light flickers under comparable conditions. Without relevant observations, a numerical estimate may be little more than a guess.

AERM should avoid statements such as “there was only a 1% chance of this happening” unless the basis for that number is clearly explained.

2. What Is a Coincidence?

A coincidence is the occurrence of two or more events at approximately the same time or in a way that appears meaningfully connected, without necessarily having a causal relationship.

For example:

• A phone rings shortly after an investigator mentions a particular person.
• A door moves while a sound is heard elsewhere.
• A device reading changes at the same time as a witness reports a sensation.
• A person notices a familiar number repeatedly during a period of heightened attention.

Some coincidences are genuinely uncommon. Others feel uncommon because people remember striking matches and forget the many times nothing matched.

The human mind is good at detecting patterns. This ability is useful, but it can also make unrelated events seem connected.

The investigative question is not simply whether two events occurred together. It is whether the relationship is more consistent with a causal explanation than with chance, shared environmental conditions, or selection after the event.

3. The Law of Large Numbers: Why More Opportunities Produce More Surprises

When an event has a small probability, it may seem unlikely on any one occasion. But if there are many opportunities for it to happen, observing it at least once becomes more likely.

Suppose an event has a probability \(p\) of occurring in one independent trial. The probability that it does not occur in that trial is \(1-p\).

For \(n\) independent trials, the probability that it never occurs is:

\[ P(\text{no occurrence}) = (1-p)^n \]

Therefore, the probability that it occurs at least once is:

\[ P(\text{at least one}) = 1-(1-p)^n \]

Example

Imagine a hypothetical event with a probability of 1% per independent opportunity.

If there is only one opportunity:

\[ P(\text{at least one}) = 1-(0.99)^1 = 0.01 \]

That is 1%.

If there are 100 independent opportunities:

\[ P(\text{at least one}) = 1-(0.99)^{100} \approx 0.634 \]

That is approximately 63.4%.

An event that seems unlikely in one trial becomes quite plausible when there are many opportunities for it to happen.

Important: This calculation assumes that trials are independent and that the probability remains the same. Real investigation observations often violate these assumptions. Ten readings taken seconds apart may be strongly related, and a device may have a changing baseline.

4. The Multiple Comparisons Problem: Finding Patterns After Looking Everywhere

Imagine that an investigation team monitors:

• 12 rooms.
• 8 sensors.
• 30-minute observation periods.
• Several different types of unusual readings.

If the team searches all the data for any unexpected pattern, there may be many opportunities to find something that looks unusual by chance.

This is known as the multiple comparisons problem.

The more comparisons or patterns examined, the greater the chance of finding at least one apparently unusual result even when no special effect exists.

A simple illustration

Suppose a researcher performs 20 independent tests, each using a 5% false-positive threshold. If all tests are actually null and independent, the probability of at least one false positive is:

\[ 1-(1-0.05)^{20} \approx 0.642 \]

That is approximately 64.2%.

This does not mean every unusual result is false. It means that a single low p-value or striking observation should be interpreted in the context of how many tests were conducted.

How investigators can reduce this problem

• Define the main question before collecting data.
• Identify the primary outcome measure in advance.
• Record all tests performed, not only the interesting ones.
• Avoid repeatedly changing thresholds until a result appears.
• Treat unplanned patterns as exploratory findings that need independent confirmation.
• Use appropriate statistical corrections when conducting multiple formal tests.

A result discovered after searching many possible patterns can be useful for generating a new hypothesis. It should not be presented as though it had been predicted in advance.

5. Base Rates: How Common Is the Event Normally?

A base rate is the frequency or probability of an event in a relevant population or setting before considering a particular piece of evidence.

Base rates matter because an event can be rare in one context but common in another.

For example, a light flickering once in a building with old electrical fittings may not be particularly surprising. The same observation in a recently tested system with stable power conditions may deserve a different level of investigation.

The important point is to compare like with like.

Ask:

• How often does this event occur in similar buildings?
• How often does the equipment produce this type of reading?
• How often do similar sounds occur when no one is investigating?
• Was the device operating within its normal range?
• Were environmental conditions comparable?

Without a suitable comparison, an investigator may mistake an ordinary event for an exceptional one.

The importance of a baseline

A baseline is a record of normal conditions against which later observations can be compared.

For example, before interpreting an unusual electromagnetic field reading, an investigator should document the background readings around electrical wiring, appliances, transformers, and other sources.

A single reading without a baseline may have very limited meaning.

The same principle applies to temperature, sound, humidity, light, and other measurements.

6. Conditional Probability: What Does the Evidence Change?

Conditional probability describes the probability of one event given that another event has occurred.

It is written as:

\[ P(A \mid B) \]

This means “the probability of A given B.”

In an investigation, a common question is:

If a witness reports a strange sound, how much does that information change the likelihood of a particular explanation?

That question is different from asking how likely the sound is under one explanation alone.

For example, suppose a sound is heard in a building. The investigator may consider two possibilities:

• \(H_1\): the sound came from an ordinary environmental source.
• \(H_2\): the sound came from an unidentified source.

The sound itself may be compatible with both possibilities. Additional evidence—such as a matching mechanical cycle, a recording from another room, or a controlled reproduction—may help distinguish them.

An event is not strong evidence for one hypothesis merely because it is possible under that hypothesis. Investigators must also consider how expected the same observation would be under competing explanations.

7. Bayesian Reasoning: Updating Beliefs as Evidence Arrives

Bayesian reasoning provides a formal way to update the relative plausibility of hypotheses when new evidence is observed.

Bayes' theorem is:

\[ P(H \mid E) = \frac{P(E \mid H)P(H)}{P(E)} \]

where:

• \(H\) is a hypothesis.
• \(E\) is the observed evidence.
• \(P(H)\) is the prior probability of the hypothesis.
• \(P(E \mid H)\) is the likelihood of observing the evidence if the hypothesis is true.
• \(P(H \mid E)\) is the updated probability after considering the evidence.
• \(P(E)\) is the overall probability of the evidence under the hypotheses being considered.

In practical terms, Bayesian reasoning asks:

1. What explanations were plausible before this observation?
2. How likely is this evidence under each explanation?
3. Does the evidence favour one explanation over the others?
4. How should confidence change after considering the evidence?

An illustrative example

Suppose a team hears a repeated knocking sound. Before assigning a cause, they consider:

• A loose pipe.
• A moving door or window.
• Footsteps or activity in another room.
• Another unidentified source.

The team then notices that the sound occurs at regular intervals and coincides with a heating system cycle.

That additional observation may increase the plausibility of a mechanical explanation. It does not prove the source, but it changes the relative support for the hypotheses.

Bayesian reasoning does not require investigators to invent precise numbers. If reliable probabilities are unavailable, the method can still be used qualitatively: identify prior possibilities, assess how well each explains the evidence, and update conclusions cautiously.

8. The Prosecutor's Fallacy and Misunderstanding Rare Events

A common statistical error is to confuse:

• The probability of observing evidence if a hypothesis is true.
• The probability that the hypothesis is true given the evidence.

These are not the same.

For example:

• “If a particular explanation were true, this observation would be unusual.”
• “This observation is unusual, therefore that explanation must be true.”

The second statement does not logically follow from the first.

There may be other explanations under which the same observation is also unusual—or explanations that make it quite ordinary.

This error is sometimes associated with the prosecutor's fallacy, especially when a small probability is treated as though it were the probability that a person or hypothesis is innocent or false.

For AERM, the lesson is straightforward: do not convert a low probability under one model into proof of an alternative explanation.

9. Correlation Does Not Establish Causation

Two measurements may change together without one causing the other.

For example, a team member reports feeling cold at the same time that a temperature sensor records a change. Several possibilities remain:

• The room temperature changed and contributed to the sensation.
• Airflow changed near the person but not near the sensor.
• The sensor was moved or affected by its surroundings.
• The person's sensation had another cause.
• Both observations were influenced by a third factor.

A correlation is a relationship in the data. Causation is a claim about why the relationship occurred.

To investigate a possible causal relationship, researchers need to consider timing, alternative causes, measurement quality, repeatability, and ideally controlled comparisons.

A single simultaneous observation rarely establishes causation.

10. The Problem of Small Samples

A small sample can produce unstable results.

Suppose an investigator records only three sessions and observes an unusual event in one of them. The proportion is:

\[ \frac{1}{3} \approx 33.3\% \]

It would be misleading to conclude that the event occurs in one-third of all investigations. Three sessions provide very little information about the underlying rate.

Larger datasets can provide more stable estimates, but only if the observations are collected consistently and the samples are relevant.

Investigators should report:

• Total number of sessions.
• Total observation time.
• Number of events recorded.
• How an event was defined.
• Missing or incomplete records.
• Whether sessions were selected or excluded.
• Whether observations were independent.

A count without an exposure measure can also be misleading. Ten events during 100 hours of monitoring are not directly comparable with ten events during 5 hours.

11. A Practical AERM Probability Framework

The following framework can help investigators evaluate an unusual event without making unsupported numerical claims.

Step 1: Define the event

Describe exactly what is being counted.

Instead of “paranormal activity occurred,” use a measurable description such as:

• A sound exceeding a pre-defined threshold was recorded.
• A sensor reading crossed a specified value.
• Two independent observers reported movement in a defined area.
• A light source changed state during a recorded interval.

The event definition should be decided before examining the results wherever possible.

Step 2: Define the observation period

Record the total monitoring time and number of opportunities for the event to occur.

For example, a 20-minute observation and a 6-hour observation should not be treated as equivalent exposure.

Step 3: Establish a baseline

Collect comparison data under ordinary conditions.

A baseline might include:

• Room temperature over time.
• Background sound levels.
• Normal electrical or electromagnetic readings.
• Typical light levels.
• Equipment noise and sensor variation.

Use appropriate instruments and record their limitations.

Step 4: List competing explanations

Write down plausible alternatives before selecting a preferred explanation.

For a sudden sound, these might include building movement, plumbing, ventilation, traffic, animals, human activity, or recording artefacts.

Step 5: Record all observations

Keep a complete record, including events that did not match the original expectation.

Do not report only the most dramatic moments.

Step 6: Examine the number of opportunities

Ask how many rooms, sensors, time intervals, recordings, or patterns were examined.

If many comparisons were made, acknowledge the increased chance of finding something unusual by chance.

Step 7: Seek independent replication

A finding becomes more useful when it can be examined again using a pre-defined method, ideally by another team or at another time.

A repeated observation is not automatically independent. The same equipment fault or environmental cycle may recur.

Step 8: State the strength and limits of the conclusion

Use language such as:

• “Observed once during the session.”
• “Repeated under similar conditions.”
• “Consistent with the proposed environmental explanation.”
• “The available data are insufficient to estimate a reliable event rate.”
• “The source remains unidentified.”

Avoid assigning a precise probability when the data do not support one.

12. Worked Example: Is a Repeated Knock Unusual?

Consider a hypothetical investigation in which a team records six knocks during a two-hour session.

At first, the team may regard this as unusual. But the count alone does not tell us whether it is rare.

The investigators should ask:

1. How often are knocks heard in the same building when no investigation is taking place?
2. Were there nearby doors, pipes, windows, or mechanical systems?
3. Did the knocks occur at regular intervals?
4. Were they recorded by more than one microphone?
5. Were the six knocks separate events or part of one continuous sequence?
6. How much time was monitored?
7. Were all sounds counted, or only those that seemed unusual?

Suppose later observations show that similar knocks occur whenever a nearby pump switches on. That would support a mechanical explanation.

If the source remains unknown, the correct conclusion is not that the knocks were paranormal. It is that the source was not identified with the available evidence.

This example demonstrates why counts need context and comparison data.

13. Common Statistical Mistakes in Paranormal Research

| Mistake | Why it is misleading | Better approach | |---|---|---| | Calling a rare event impossible | Low probability is not zero probability | Describe the estimated probability and its assumptions | | Reporting only unusual results | Hides the number of opportunities and null observations | Maintain a complete investigation log | | Searching many patterns and reporting one | Ignores multiple comparisons | Predefine primary outcomes and disclose exploratory analyses | | Treating a coincidence as causation | Simultaneous events may share no causal link | Test alternative explanations and timing | | Using a percentage from a tiny sample | Estimates may be highly unstable | Report counts, exposure, and uncertainty | | Assigning arbitrary probabilities | Creates false precision | Use measured baseline data or qualitative reasoning | | Treating a p-value as the probability a hypothesis is true | Misinterprets what the statistic means | Explain the model and the question the test addresses | | Ignoring dependence between observations | Repeated readings may not be independent | Examine the data structure and measurement process |

14. AERM Research Perspective

Probability is not a tool for explaining away unusual events. It is a tool for evaluating how much weight an observation deserves.

A coincidence can be striking. A pattern can be interesting. A sensor reading can be unusual. But the investigator must ask how often the event could occur under ordinary conditions, how many opportunities existed for it to happen, and whether the observation was selected after looking through a large amount of data.

AERM should prefer complete records, pre-defined event criteria, suitable baselines, transparent calculations, and independent replication over dramatic claims based on isolated moments.

No anomaly is considered paranormal until environmental, psychological, technical, and scientific explanations have been systematically evaluated.

The purpose of statistical reasoning is not to force every event into a convenient explanation. It is to prevent uncertainty from being mistaken for certainty.

Conclusion

Coincidences happen, including coincidences that seem extraordinarily meaningful. Probability helps investigators understand why a rare event can occur, why more opportunities produce more apparent surprises, and why searching through large datasets can reveal patterns that do not replicate.

Base rates provide context. Conditional probability and Bayesian reasoning help compare explanations. Multiple-comparison awareness prevents selective reporting. Careful measurement and complete records make statistical conclusions more defensible.

For AERM, the most important habit is to define the event before interpreting it, record every relevant observation, compare results with ordinary conditions, and be honest about uncertainty.

An event does not become paranormal simply because its explanation is not immediately obvious. Sometimes the right conclusion is that more data are needed.

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References and Further Reading

These sources provide foundations for probability, statistical reasoning, Bayesian inference, and research design. Verify publication details and use the original texts when preparing formal academic work.

1. Gigerenzer, G. (2002). Calculated Risks: How to Know When Numbers Deceive You. Simon & Schuster.
2. Good, I. J. (1950). Probability and the Weighing of Evidence. Charles Griffin.
3. Hacking, I. (2001). An Introduction to Probability and Inductive Logic. Cambridge University Press.
4. Kahneman, D. (2011). Thinking, Fast and Slow. Farrar, Straus and Giroux.
5. Mayo, D. G. (2018). Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars. Cambridge University Press.
6. McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and Stan (2nd edition). CRC Press.
7. Wasserstein, R. L., & Lazar, N. A. (2016). The ASA's statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133.

About AERM

Anomalous Evidence Research & Media (AERM) is an independent research and media initiative focused on the systematic documentation and evaluation of unusual reports. Its approach combines environmental assessment, witness documentation, scientific reasoning, and evidence analysis.

AERM does not treat a coincidence, rare measurement, or unexplained event as proof of a paranormal cause. Each report should be evaluated against plausible environmental, psychological, technical, and statistical explanations.

Educational disclaimer: This article is for general educational and research-awareness purposes. It is not a substitute for professional statistical analysis or a formal scientific study. Numerical examples are illustrative and depend on the assumptions stated. Do not report calculated probabilities without documenting the data, model, and limitations.