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We constructed two datasets, respectively a set of standard exercises and a set of counterintuitive exercises, designed to trigger heuristic reasoning, and evaluated 8 state-of-the-art models, each tested with and without Chain-of-Thought prompting. Models achieve an average accuracy of 0.96 on standard problems but only 0.59 on counterintuitive ones. We further provide empirical evidence of token bias: performance drops by over 20% when canonical formulations are replaced by disguised variants. Embedding misleading suggestions in the prompt reduces performance by up to 34%, with no model proving immune. Taken together, the reported findings suggest that current LLMs are not yet genuine probabilistic reasoners, despite their success in advanced mathematical problems.","title":"How reliable are LLMs when it comes to playing dice?","url":"https://arxiv.org/abs/2606.07515","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.07515v2 Announce Type: replace-cross \nAbstract: We investigate the probabilistic reasoning capabilities of large language models through a controlled benchmarking study on discrete probability problems. We constructed two datasets, respectively a set of standard exercises and a set of counterintuitive exercises, designed to trigger heuristic reasoning, and evaluated 8 state-of-the-art models, each tested with and without Chain-of-Thought prompting. Models achieve an average accuracy of 0.96 on standard problems but only 0.59 on counterintuitive ones. We further provide empirical evidence of token bias: performance drops by over 20% when canonical formulations are replaced by disguised variants. Embedding misleading suggestions in the prompt reduces performance by up to 34%, with no model proving immune. Taken together, the reported findings suggest that current LLMs are not yet genuine probabilistic reasoners, despite their success in advanced mathematical problems.","title":"How reliable are LLMs when it comes to playing dice?","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-12T04:43:44Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.07515"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:362ced872e9960b236f25162319970501a6548b68c134df6aab9d30760d25b8580340378c7048ff6ff485c9cbcfd9db05473c3e477b889dfc0425d86ca552f0b","signer":"crovia.substrate","subject":{"observed_at":"2026-06-12T04:43:44Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.07515"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"033fe72c1fce80308f92bf75a4185f64206a6dc27243d66db7ae68a996f74f3f","leaf_index":229906,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"3c5a4c03dabff0d4f394b0143beee49882a8fa8dbfa918190e6984eb05654cc5","side":"right"},{"sibling":"a3d78ead8fe2b523d4e2722de04d8b2b2f77433e1fc00ba1893fa1dd5c2d94b1","side":"left"},{"sibling":"f70c519745a5afaf6250765ca579ace839ab68ddf8dffc53bd2357adea6aaacc","side":"right"},{"sibling":"b8feb328f9bd410b91a4be251f06db007788201d9ce59ce8c4c04d2fb841e147","side":"right"},{"sibling":"f39b5e7740111dc23b9c244a02db8d972acdcd5b405c4169f5f3e880fa2baac6","side":"left"},{"sibling":"bf0e631fe6e0bc7e2f7b67b2bac99bc045b976d64492dd645f4d30340e7d6c02","side":"right"},{"sibling":"6bde248411f8ec3173e97ec3895054895c25f208bf85a22fd09c8fdf4911671b","side":"right"},{"sibling":"bc1002bb7e3da047b8a7c8f3990db61fb56edf499868605dc0c31aa0dee387b7","side":"right"},{"sibling":"14c50c43949e1ad41f149ffea691627d3f715c5861766c693b9fbac9d03b0d90","side":"right"},{"sibling":"74897e850164dddc689c3c65b33f9bae0268ab0bf429867a4e193d9b9b685040","side":"left"},{"sibling":"bde25d7e94e64717e426a97f6fcb4907e92b5c61fc89d92d7e0947a2249c3f6b","side":"right"},{"sibling":"d10d772a4984cae00e65ab24af21d1d260e475bbaae3f17871859eb705bd3999","side":"right"},{"sibling":"e79159853f2f35ddae8e3247d515e433c534277b287d65bbd77ae989aa4992fa","side":"right"},{"sibling":"3054319f1840cce0eaaf0bc4b1ae38e5bf8b6210927d924a750775cc7d77cca6","side":"right"},{"sibling":"0fd8b5059f279c4a4a6688de2472fdbc25543fee183df19b21dacca880354cff","side":"right"},{"sibling":"a04392fb9f2a3a620840e3b3fecd93d12e6d224e481c162389d8ae64e7194599","side":"left"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":232015,"merkle_root":"62bfb7809bb55667ad7eeebdb48267b9b2c1ee89bb808ea1e6d3a814a15aa402","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260617T133701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-17T13:39:59Z","sig_algorithm":"ed25519","signature":"930a3563c643cc7518d048b12a1f5392a96a5edce49533ba0a56f11b6cc69319bb1c800aacc46b52a6941c4d05dae255a45cac757d6093c971b96852c24f140e","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_0abbe49167041b31d4c19e3e831c1948f52c6c683d43fcffd55fccbf1a0df900"}}