Observation and Reporting in Conditional Probability — One Ace versus the Ace of Hearts

In practice, decision making rarely depends on the full information that was originally observed¹. Instead, decisions are often based only on the information that was eventually reported. This distinction is fundamental.

An observer may possess richer latent information than the information communicated through the reporting protocol. Consequently, two situations that appear equivalent at the reporting level may correspond to very different informational structures.

Conditional probability is usually introduced theoretically through abstract events and sigma-algebras. However, in practical applications, conditioning frequently depends on:

• the observation mechanism;
• the reporting protocol;
• and the granularity of the transmitted information.

This issue appears naturally in many applied domains, including actuarial reserving and IBNR modelling, where the observed reality and the reported information may differ substantially.

From a statistical perspective, the objective is to construct the richest admissible information set generated by the observation. In Hilbert-space terms, the conditional expectation $ E[X \mid Y]$ is the orthogonal projection of X onto the space of all measurable functions g(Y). Consequently, inference should exploit the entire $ \sigma-\text{algebra}$ generated by the observation, including any information carried by the reporting protocol itself. Treating the protocol as non-existent amounts to projecting onto a smaller information space, thereby discarding potentially relevant information.

Problem formulation Link to heading

Consider a standard deck of 52 playing cards. Two cards are drawn uniformly at random. Suppose different observers obtain partial visual information² about one of the two cards and communicate what they believe they observed. We compare the following situations:

1. Adi states: «I see an ace.»
2. Alex states: «I see the ace of hearts.»

Suppose somebody offers the following bet:

If the hand contains two aces, you win at odds of 20:1 .Would you accept the bet under each reporting protocol?

The answer depends not only on the numerical conditional probability itself, but also on how the observation and reporting mechanisms are interpreted. In particular, an important distinction emerges between:

• information that was actually observed;
• and information that was merely reported.

Baseline probability Link to heading

Without any information:

$$ P(2A)=\frac{\binom{4}{2}}{\binom{52}{2}}=\frac{1}{221}\tag{prior} $$

Thus, drawing two aces is initially very unlikely.

Case 1 — we have an ace Link to heading

Define the event:

$$ B={\text{the hand contains at least one ace}} $$

The compatible hands are:

• exactly one ace:

$$ 4\times48=192 $$

• two aces:

$$ \binom{4}{2}=6 $$

Therefore:

$$ |B|=192+6=198 $$

Among these 198 hands, only 6 contain two aces. Hence:

Case 2 — we have the ace of hearts A♥ Link to heading

We now condition on a more specific informational structure. To remain consistent with the previously defined sample space of 198 hands (hands containing at least one ace), we re-express the counting within this universe.

Within these 198 admissible hands, we can consider a symmetric decomposition. If we fix one specific ace (the ace of hearts), then the second card can be any of the remaining 51 cards. This generates 51 hands containing the ace of hearts.

Thus, within the full structure:

• for each of the 4 aces there are 51 associated hands;
• this gives a total of: 4 × 51 = 204

However, this construction double-counts hands containing two aces, since each such hand appears in two different ace-groups. There are $ \binom{4}{2} = 6$ such double-ace hands. Correcting for this overlap: $ 4 × 51 − 6 = 198$, which recovers the original conditioned universe.

Now restricting to the ace of hearts³, the relevant subset is exactly 51 hands, among which 3 contain a second ace. Therefore:

Why do the probabilities differ in the two cases?

The usual explanation says: ace of hearts is more precise information than an ace. This explanation is correct but incomplete. The deeper issue is the information protocol.

Triggered versus induced information Link to heading

The very fact that a particular message was chosen may itself reveal information about the underlying observation process. In such situations, the reported event does not merely trigger conditioning; it induces an additional informational structure.

This distinction can be summarized as follows:

This highlights how conditioning on events requires attention to how the message was generated, not just what it reports.

Competitive advantage Link to heading

An important conceptual point emerges here. The fact that we cannot obtain absolute certainty does not imply that all hypotheses are equivalent — and recognizing this asymmetry is precisely what gives rise to competitive advantage.

If Adi behaves as a perfectly rational agent in the sense of E. T. Jaynes, then the wording carries no additional information beyond the literal event being communicated. The message is assumed to be generated according to a known and internally consistent reporting mechanism.

Under such assumptions, no extra inference should be extracted from the linguistic choice itself. However, real observers are not ideal Bayesian machines.

In practice, the agents may contain additional latent information regarding:

• observational precision;
• communicative competence;
• selective reporting;
• or the observer’s understanding of what information is relevant.

A skilled actuary cannot afford to leave this information on the table.

Observer credibility Link to heading

The previous calculations correspond to two extreme interpretations (coarse + fine). In practice, neither extreme is entirely realistic.

Let

$$ C={\text{report }R\text{ correlates 1:1 with a specific ace}} $$

and let $ c=P(C\mid R)$ denote the degree of plausibility⁴ that the report genuinely corresponds to a one-to-one identification of a specific ace.

Negating C does not mean that the observation identifies a different ace. Rather, it means that the assumed one-to-one relationship between the observation and a specific ace does not exist. As a limiting case, we model this absence of correspondence by inferring that the report carries no information relevant to the event 2A. Consequently,

$$ P(2A\mid \neg C)=P(2A). $$ Using the law of total probability, the final posterior is:

P(2A \mid R) = \underbrace{P(2A \mid C)}_{\text{fixed}}\cdot P(C \mid R) \;+\; \underbrace{P(2A \mid \neg C)}_{\text{fixed}} \cdot P(\neg C \mid R)

Substituting the two limiting probabilities yields

$$ P(2A\mid R) =\frac{1}{221}+c(\frac{3}{51}-\frac{1}{221}).\tag{Bühlmann} $$

Thus:

• $ c=0$ recovers the prior probability 1/221;
• $ c=1$ recovers the fully specified observation 3/51 ⁵;
• $ c=\frac{47}{99}\approx 47.5\%$ classical answer 1/33 is recovered.
• intermediate values represent partial credibility assigned to the observer.

This transforms the problem from a purely combinatorial exercise into an inference problem involving both the cards and the credibility of the reporting mechanism.

From a decision-theoretic perspective, the relevant quantity is no longer the conditional probability itself, but the credibility threshold required to justify a particular action.

For example, at odds of 20:1, the bet becomes favorable whenever⁶,

$$ c>\frac{\frac1{21}-\frac1{221}}{\frac1{17}-\frac1{221}}\approx79.4\%. $$

The cards are the same for everyone. Competitive advantage arises precisely from estimating quantities such as c more accurately than others. Within this framework, if the available evidence makes $ c \gtrsim 79.4\% $⁷ plausible, accepting the risk becomes the rational decision.

Connection with IBNR Link to heading

The example is closely related to IBNR reasoning. In many actuarial settings:

• the underlying reality already exists;
• the reporting system only reveals a partial projection (RBNS + Payments);
• the reporting protocol is induced by the system itself.

If reported data are treated as the complete informational structure, additional segmentation may become artificial⁸. However, once the latent observation mechanism is acknowledged, many apparent refinements no longer introduce genuinely new information.

It concerns whether observed data are interpreted as:

• complete information;
• or compressed information originating from a richer latent structure.

Conclusion Link to heading

Conditional probability depends not only on the observed event itself. It also depends critically on:

• the observation protocol;
• the granularity of the reported information;
• and the assumed mechanism generating the message.

This distinction appears repeatedly in applied statistics, reserving theory, and inference under incomplete reporting.

This mindset defines the core of Actuarial Mathematics, especially through credibility theory. You don’t need to master every detail — you just need the right actuary to build the right model.

@online{Cornaciu2026Aces,
  author   = {Cornaciu, Valentin},
  orcid    = {0000-0001-9239-7145},
  title    = {Observation and Reporting in Conditional Probability — One Ace versus the Ace of Hearts},
  year     = {2026},
  date     = {2026-06-02},
  url      = {https://rcor.ro/posts/2026-05-24-observation-and-reporting-in-conditional-probability-one-ace-versus-the-ace-of-hearts/},
  abstract = {This article explores the distinction between observed information
  and reported information in conditional probability. Using the classical 
  «one ace» versus «ace of hearts» example, the paper argues that conditional
  probabilities depend not only on abstract events, but also on the underlying
  information protocol and reporting mechanism. The discussion further connects
  these ideas with latent information, coarse-graining, and incomplete reporting
  frameworks encountered in actuarial reserving and IBNR modelling.}
}