A problem with three correct answers. Which, how, and when does a model choose?

TL;DR: A look at how models (3 sizes of the Qwen 2.5 model family) behave when posed with Bertrand's paradox. 18 phrasings, up to 30 answers each. For each, we check which of the three famous responses it outputs, whether it explains the method or recalls it, and when the method choice shows up in the model's internals.

When models are posed with an ambiguous question, it is very useful to know whether we can identify their choice through the chain of thought transcript alone. We asked whether the method choice is represented in the model's activation before it is written down, and when.

Findings:

• 0/761 answers state that the problem is ambiguous. The model settles on an approach and provides the final answer as if it were the only possibility.
• For the 7B, the phrasing nudges towards a final number (no radius given: ⅓, radius given: ½). For the smaller models, the distribution of final numbers remains consistent across phrasings.
• Most answers explain their reasoning (85% for 7B, 55% for 3B). The famous ⅓ is the one that gets recalled (plainly stated) most often, and the reasoning builds up to it while the model knows what it is supposed to reach (75% of ⅓s for 3B, 22% for 7B)
• The approach the model settles on becomes distinguishable only 5-10 tokens before the pivotal word (the one that nails the method down: "central angle", "perpendicular distance", and other similar ones). The reading at 30 tokens before it does not add anything else that the phrasing and the text up until that point didn't already provide, but a probe detects a 5-10 token build-up (from 53-57% to 79% at the exact position of the pivotal token).

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

This is Bertrand's paradox.

Based on how we draw the chord, we have three approaches with three corresponding answers: ⅓, ½, ¼. All of these are correct.

Setup

Models

We test three Qwen2.5 Instruct models: 1.5B, 3B, and 7B (in 4-bit). We sent the prompt that describes the problem as a single user message, with no added instructions or examples (the Qwen chat template with its default system line).

Rephrasing the problem

Because this is a famous problem, it is very likely that the models are already familiar with it and can recall solutions from memory. We went through the experiment with this kept in mind (check How), and one way to mitigate this was to have multiple prompt phrasings (ways to present the problem): besides Bertrand's paradox standard phrasing, we have 17 rewordings. They vary by:

• radius (1, 2, 5, or not given)
• comparison length: the triangle side, (the length of the triangle's side), or the already computed number (for example, )
• sentence order
• what the verb of the question is.

All 18 phrasings keep the “chord” wording, but none mention how it could be drawn and understood.

Two kinds of prompts

We have two kinds of prompts, each with its own purpose:

• Neutral prompts: a simple phrasing. The model has to decide for itself how it draws the chord.
• Specified prompts: a simple phrasing with an additional sentence attached to it, with the purpose of telling the model how to draw the chord (for example: In this problem, "at random" means picking two points independently and uniformly on the circle.)

Each approach is explained in 8 different ways (descriptions): for the ⅓ approach, we could tell the model to either randomly pick two points on the circle or to fix one point on the circle and place the other randomly. This gives us a total of 24 descriptions.

The main experiment is run only on neutral prompts, but the specified prompts are useful in two other ways. Firstly, as a check: if a model can’t reach ⅓ even when told how to approach the problem, we can't trust its judgements for its final numbers (this was the case for 1.5B, as it only matched the approach’s corresponding number in ¼ of its answers). Additionally, we train the probes we use in a , because specifying the approach can make us much more certain that it is the one the model actually uses.

Answers

For each neutral prompt, we collected 30 answers (10 answers for 1.5B) for each phrasing and model by posing them with the same prompt under the following settings:

• Temperature 0.7
• Top-p 0.95
• No repetition penalty
• Up to 1024 tokens

For specified prompts, we set the sampling off and collect a single answer per prompt. We do this because we only need to know whether the model follows the instruction, as opposed to seeing the spread for the neutral prompts.

In total, we get 540 (18 phrasings x 30 answer samples, or 180 for 1.5B) answers for the neutral prompts, and 432 (18 phrasings x 24 descriptions) answers for the specified prompts per model.

Labels

We label each answer by its final number: ⅓, ½, ¼, or other. The last label includes answers that never output a final number or meet the upper token limit.

To extract this output, if the model states the final answer in a LaTeX box (for example, \boxed{\frac{1}{3}} for ⅓), we use that (this happens in 94% of cases for 3B and 7B). When it doesn’t, we search for phrasings like “the probability is” and take the number that follows, and otherwise, we resort to picking the last number that appears in the text.

We label by the number because the models state their method in only a bit under half of their answers (42% of the 3B's answers and 29% of the 7B's answers say how they choose the chord), and when they do, we store that information as an extra check (we elaborate on this later). Labeling by the method would only keep a minority of answers that happen to specify it, and we wanted to analyze the general behaviours of these models when posed with Bertrand’s paradox.

Which?

Which one of the three numbers does the model reach?

The ambiguity slips under the radar

We checked 761 analyzed answers (the ones that reach ⅓, ½, or ¼: 361 from 3B, 400 from 7B) to see whether they mention the paradox, the ambiguity of the problem, or the fact that the answer depends on how the chord is drawn. It never happened, so as a second check, we conducted a keyword search (we searched for the following sequences: “paradox”, “ambiguous”, “depends on how”, and similar others) on all 1260 answers. It resulted in 21 hits, but all of them were sentences along the lines of "the chord's length depends on its distance from the center”, so they did not refer to the ambiguity.

Instead, after the answer text reaches the chord part, it simply describes it as if it were the only possibility and moves on. However, even for the 7B model, only 29% of answers explicitly state in text how the chord is picked, and they view it as the method, not a choice, and the others simply don't mention it at all.

The phrasing of the problem is relevant only for the biggest model

It is natural to assume that the phrasing serves as a hint and steers the model towards a final number. Well, does it?

We test this in two ways:

• Across phrasings: we organize the 540 answers in a table (rows for phrasings, one possible output for the columns) and check if the rows vary more than they would look like after a random shuffle. To do it, we randomly assign output numbers across phrasings 5000 times, and we check how uneven the table is compared to the original one (41% of the time for 1.5B, 10% for 3B, and never for 7B)
• Within a phrasing: we look at a phrasing’s 30 answers and measure how often they agree on a final output. We compare the average to what we would get if each of the 30 answers were randomly selected from the answer distribution (49% versus 48% for 3B, and 61% against 51% for 7B)

For 1.5B and 3B, the answer distribution remains roughly the same, and we did not detect any phrasing effect. For 7B, however, we found that the final number the model settles on is steered by two phrasing characteristics: whether the radius is given and how the comparison length is presented.

Phrasing 6: Consider a circle. A chord of the circle is drawn at random. What is the chance that it is longer than the side of an equilateral triangle inscribed in the circle?

No radius, the size of the triangle’s side as the comparison length => ⅓ in 18 of 20 answers that reached a final number

Phrasing 13: Consider a circle of radius 1. A chord of the circle is drawn at random. Find the probability that it is longer than √3 times the radius.

Radius given, comparison length as √3 times the radius => ½ in 17 of 26 answers that reached a final number

Generally, this is the pattern we identified:

• The radius is not given: mostly ⅓
• The triangle’s side as the comparison length: mostly ⅓
• Given radius + the comparison length written as a number or as r: leans toward ½
• ¼ is almost never in the majority, and it is absent in four phrasings

Of course, we are aware that the final number is only the end of the reasoning and problem-solving process: an answer that describes the chord by the angle it spans at the center results in ⅓, but one that is based on its distance from the center reaches ½. The choice is actually which description of the chord the model settles on, and the final number follows.

The answers from the specified prompts

We consider that an answer aligns with the specified approach if the final numbers match. This happens 25% of the time for 1.5B, 46% for 3B, and 71% for 7B. For the latter, it gets ⅓ right 63% of the time, ½ in 94% of cases, and ¼ in 56%. Due to its weak performance, we didn't include the 1.5B model in subsequent analyses.

So our understanding is that even though phrasing influences the odds, it doesn't directly nail a final number down, as the same phrasing for 7B still produces ½ and ¼ answers in a non-negligible manner.

How?

Does the model recognize the problem and recall a correct answer, or does it reach a final number through actual reasoning?

Because Bertrand’s problem appears in many introductions to probability, it is very likely that models are familiar with it from training, especially with its most famous result, ⅓. If we see this output, how do we check whether it worked it out while solving the problem, or built its supporting reasoning aware of the result it was supposed to reach? The two look the same in the answer’s text, but the model may have operated differently.

We classify each answer that resulted in ½, ⅓, or ¼ as follows (because they are only text properties, we can use them to only guess the model's true process, not assert it):

• explained: there are written steps that lead to the final number (we are lenient with the small slips and mistakes that might happen along the way, as long as most of the reasoning and the answer are correct)
• unexplained:
  • the model arrives at a different number (for example, ⅙ or ⅔) but writes ⅓ in the LaTeX box anyway
  • the reasoning has the right shape, but its components are wrong (for example, "the chord is long when the central angle is 120° or more, so the probability is 120/360 = ⅓")
  • the number is simply stated on its own, with no computation or formula to account for it

The answers were classified by a Claude Code session. Regarding the explained answers, each picked one concept to "randomize": the angle the two random points on the circle form at the center, the distance between their midpoint and the center, or where the midpoint falls.

The ⅓ answer is the most famous one, and this is reflected in the explained/unexplained split, as the unexplained ⅓ answers account for 91/121 answers for 3B and 40/182 for 7B.

The unexplained ⅓ answers might really just be a recall, as they are most commonly associated with the standard answer and the standard phrasing. Of course, an explained answer might still be recalled, but the model provides the reasoning with the final answer it is supposed to reach in mind (the model operates differently in ).

The true choice is how the model picks to interpret the chord, but if this is the case, could the choice be visible before the moment it appears in writing?

When?

Can the choice be read from the model's activations before it appears in text?

To check the model's internals while it is solving the problem it was posed with, we use probes: linear classifiers looking at the model's internal activations at one layer, trained to predict the label of the resulting answer. To be able to interpret the probe's output, we compare it to the outputs of three other references:

• simple chance: 50% for 2 possible choices, 33% for 3 possible choices
• shuffled labels: the same probe, but trained after the labels got shuffled and randomly reassigned. They get shuffled multiple times, and we look at the 95th percentile. This can also be classified into:
  • global: the labels get shuffled across all answers
  • within a phrasing: the labels get shuffled across the answers corresponding to a certain phrasing
• words-only: instead of a probe, we use a simple text classifier. This one is only trained on the same text the probe was shown (the prompt and the answer up to that position), but it has no access to the model's activation, so it can't use them for its prediction (if the probe doesn't beat it, it is only reading words, not the model's internal activations)

Where we look

For specified prompts, we check right at the last token of the prompt. The choice is readable, as a probe scored 96% on unseen phrasings with varied descriptions (against 33% for chance, 33% for shuffled labels, and 76% for words-only). Besides showing that the model represents the specified approach, it also serves as a check for the pipeline and for the fact that an approach can be represented in the model's internals.

When the model makes its own choices, we look at the "pivotal" word (the first word that names the concept that most certainly leads to the final answer: "perpendicular distance", "angle", "midpoint"), and we start walking backwards from it. When we look at the very start of the answer, the probe scored 53%, but it also scored 45-48% shuffled, and 48% words-only, so the probe was only exploiting the phrasing and the written words, not detecting an earlier choice. We decided to probe the pivotal token instead.

The model's choice and the pivotal word go together

For the explained answers, this was how the distance from the pivotal word was reflected by the probe's output:

• 30 tokens before: 40% (41% shuffled, 42% words-only)
• 10 tokens before: 53% (46% shuffled, 49% words-only)
• 5 tokens before: 57% (47% shuffled, 48% words-only)
• 1 token before: 75% (48% shuffled, 59% words-only)
• the token itself: 79% (52% shuffled, 72% words-only)

The decision cannot be detected long before the pivotal word, but there is a build-up of 5-10 tokens to it (this aligns with the notion of a model looking at the general sentence the pivotal token belongs to).

The explained/unexplained split

We investigated how and whether a model reaches the famous ⅓ answer or not with the help of a binary probe (so the chance is 50% this time).

• explained answers:
  • 30 tokens before: 63% (61% shuffled within phrasings)
  • 1 token before: 92%
  • the token itself: 98% (94% words-only, so the word steers the final number)
• unexplained answers, which turned out as not readable:
  • 1 token before: 38%
  • the token itself: 56% (57% shuffled, 47% words-only)

Wrapping up

Conclusion

The model never treats Bertrand's problem as ambiguous, and while the phrasing of it does nudge the final number in a certain direction (for example, ½ is more likely if the radius is given, and ⅓ otherwise), the choice is made when the model writes down the principal concept the approach relies on. Additionally, the famous, standard phrasing triggers a recall of the famous ⅓ answer. Even though we detect a few-token build-up, we cannot attribute it to a decision made in advance, rather, the model places the pivotal token in the context of the sentence it belongs to. When the model does settle on an approach, it outputs its matching final response and does not revisit other methods.

Limitations

All our analyses are based on a single model family, and while the probes were trained on a sweep of the model's layers, we report them only at a single one. We did not test any reasoning model, whose characteristic deeper exploration could have changed how the model treats and tries to solve the problem, and the findings could have been more diverse.

Where do these results fit in the bigger picture

It is already known that models will answer ambiguous questions without flagging ambiguity. Because Bertrand's paradox solutions generally include one construction and one answer, the model probably learned this structure. The text the model writes down before the pivotal position can be matched with any of the three approaches, so which one it settles on mostly comes down to sampling, even if phrasing characteristics influence the odds.

Regarding chain of thought monitoring, it is good to know that the written text represents the choice at one identifiable position. However, the transcript does not directly spell it out as a choice or pivot, so we must derive which words are decisive ourselves depending on the task the model needs to complete. The problem here is not concealed reasoning, but a transcript structure we do not have a thorough understanding of.

Initially, we set out to answer whether the choice could be steered through the model’s activations, but we did not run this experiment, because the “when” question section already predicts what it would have meant. Before the pivotal word, there is not much to push on, and when it does appear in text, the choice can be read out through text by itself (79% for the probe against 72% for words-only), with little need to work with the model’s internals. Steering towards an approach would only imply pushing the model to output certain tokens, which we can do much more simply by just editing the text. For the explained answers, the reasoning is never hidden, and the steps are laid out in text as the model naturally arrives at them.

GitHub repo: https://github.com/star2vec/bertrand