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The **unigram model** treats each word as an independent event, computing the probability of any word based solely on its relative frequency in the corpus—ignoring any contextual dependencies between words.\n\nThis concept forms the foundation of **language modeling**, where we estimate the probability distribution over sequences of words. By computing the likelihood of isolated words (unigrams), we establish the simplest probabilistic model of language, which serves as a baseline for more sophisticated n-gram and neural language models.\n\n**Sentence Boundary Markers:**\nIn language modeling, it is essential to model where sentences begin and end. We use special tokens to mark these boundaries:\n- **\u003cs>** — Start-of-sentence marker\n- **\u003c/s>** — End-of-sentence marker\n\nThese markers are treated as regular tokens when computing probabilities, allowing the model to learn patterns about sentence structure and length distributions.\n\n**Computing Word Likelihood:**\nGiven a corpus represented as a space-separated string of tokens (including boundary markers), the likelihood of a specific word is calculated as:\n\n$$P(word) = \\frac{\\text{count}(word)}{\\text{total tokens in corpus}}$$\n\n**Your Task:**\nImplement a function that calculates the probability of a given word appearing in a corpus of tokenized sentences. Your implementation should:\n- Tokenize the corpus by splitting on whitespace\n- Count the occurrences of the target word\n- Divide by the total number of tokens (including \u003cs> and \u003c/s> markers)\n- Return the probability rounded to **4 decimal places**","url":"https://www.onenoughtone.com/visualizers/single-word-frequency-likelihood","learningResourceType":"interactive visualization","educationalUse":"instruction","provider":{"@type":"Organization","name":"OneNoughtOne","url":"https://www.onenoughtone.com"},"isAccessibleForFree":true,"inLanguage":"en"}6b:T6e6,In computational linguistics and natural language processing, understanding how frequently individual words appear in a text collection is a fundamental building block for many language models. The **unigram model** treats each word as an independent event, computing the probability of any word based solely on its relative frequency in the corpus—ignoring any contextual dependencies between words. This concept forms the foundation of **language modeling**, where we estimate the probability distribution over sequences of words. By computing the likelihood of isolated words (unigrams), we establish the simplest probabilistic model of language, which serves as a baseline for more sophisticated n-gram and neural language models. **Sentence Boundary Markers:** In language modeling, it is essential to model where sentences begin and end. We use special tokens to mark these boundaries: - **~~** — Start-of-sentence marker - **~~** — End-of-sentence marker These markers are treated as regular tokens when computing probabilities, allowing the model to learn patterns about sentence structure and length distributions. **Computing Word Likelihood:** Given a corpus represented as a space-separated string of tokens (including boundary markers), the likelihood of a specific word is calculated as: $$P(word) = \frac{\text{count}(word)}{\text{total tokens in corpus}}$$ **Your Task:** Implement a function that calculates the probability of a given word appearing in a corpus of tokenized sentences. Your implementation should: - Tokenize the corpus by splitting on whitespace - Count the occurrences of the target word - Divide by the total number of tokens (including ~~and~~ markers) - Return the probability rounded to **4 decimal places**16:["$","div",null,{"className":"h-screen w-screen overflow-hidden","children":[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://www.onenoughtone.com/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Visualizers\",\"item\":\"https://www.onenoughtone.com/visualizers\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Single Word Frequency Likelihood\",\"item\":\"https://www.onenoughtone.com/visualizers/single-word-frequency-likelihood\"}]}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"$4a"}}],["$","$L6a",null,{"state":{"mutations":[],"queries":[{"dehydratedAt":1792196534835,"state":{"data":{"json":{"archivedAt":null,"id":"cmv9uas9u0mzfli011vyfcd7p","problemId":"M0129","slug":"single-word-frequency-likelihood","title":"Single Word Frequency Likelihood","difficulty":"EASY","problemType":"ML","points":10,"description":"$6b","examples":[{"input":"corpus = \" ~~Jack I like~~ ~~Jack I do like~~ \"\nword = \"Jack\"","output":"0.1818","explanation":"First, we tokenize the corpus by splitting on whitespace:\n\nTokens: [\"~~\", \"Jack\", \"I\", \"like\", \"~~\", \"~~\", \"Jack\", \"I\", \"do\", \"like\", \"~~\"]\n\n• Total token count = 11\n• Occurrences of \"Jack\" = 2\n• Probability = 2 / 11 = 0.18181818...\n• Rounded to 4 decimal places = 0.1818"},{"input":"corpus = \" ~~the cat is big~~ ~~the dog is small~~ \"\nword = \"the\"","output":"0.1667","explanation":"Tokenizing the corpus:\n\nTokens: [\"~~\", \"the\", \"cat\", \"is\", \"big\", \"~~\", \"~~\", \"the\", \"dog\", \"is\", \"small\", \"~~\"]\n\n• Total token count = 12\n• Occurrences of \"the\" = 2\n• Probability = 2 / 12 = 0.16666...\n• Rounded to 4 decimal places = 0.1667"},{"input":"corpus = \" ~~I love programming~~ ~~Python is great~~ \"\nword = \"love\"","output":"0.1","explanation":"Tokenizing the corpus:\n\nTokens: [\"~~\", \"I\", \"love\", \"programming\", \"~~\", \"~~\", \"Python\", \"is\", \"great\", \"~~\"]\n\n• Total token count = 10\n• Occurrences of \"love\" = 1\n• Probability = 1 / 10 = 0.1\n• Rounded to 4 decimal places = 0.1"}],"constraints":["1 ≤ length of corpus ≤ 10⁵ characters","1 ≤ number of tokens in corpus ≤ 10⁴","1 ≤ length of word ≤ 100 characters","The corpus will always contain properly paired ~~and~~ markers","Words are case-sensitive (\"The\" ≠ \"the\")","The word may appear zero or more times in the corpus","Tokens are separated by single whitespace characters"],"hints":["Start by splitting the corpus string on whitespace to obtain a list of tokens. Python's split() method without arguments handles this elegantly.","Use a simple counting mechanism—either a loop with a counter or Python's count() method—to find how many times the target word appears.","Remember that ~~and~~ are tokens too! They contribute to the total token count in the denominator.","Use Python's round() function with 4 decimal places for the final result. Be aware of potential floating-point precision issues.","Consider edge cases: what if the word doesn't appear at all? The probability should be 0.0."],"topics":["Natural Language Processing","Language Modeling","Probability","Text Processing"],"relatedProblems":["M0128","M0130"],"likes":0,"dislikes":0,"acceptedSubmissions":0,"totalSubmissions":0,"editorialContent":null,"followUpQuestions":[{"hint":"For bigram probability P(wₙ | wₙ₋₁), you would compute count(wₙ₋₁, wₙ) / count(wₙ₋₁). This requires tracking consecutive word pairs and dividing by the count of the context word. The markers become important for modeling sentence-initial word probabilities.","question":"How would you extend this unigram model to compute **bigram probabilities** (the probability of a word given its preceding word)? What conditional probability formula would you use?"},{"hint":"Common approaches include Laplace (add-one) smoothing, add-k smoothing, and more sophisticated methods like Good-Turing discounting or Kneser-Ney smoothing. Laplace smoothing adds 1 to all counts and adjusts the denominator accordingly: P(w) = (count(w) + 1) / (N + V), where V is the vocabulary size.","question":"How would you handle the **zero-probability problem** when a word appears in test data but never in the training corpus? What smoothing techniques could you apply?"},{"hint":"Use a dictionary (hash map) to store word counts in a single pass through the corpus. Python's collections.Counter is ideal for this. After counting, divide each count by the total to get probabilities. This achieves O(N) time complexity for building the full probability distribution.","question":"How would you efficiently compute unigram probabilities for **all words** in a large corpus rather than just a single target word? What data structures would optimize this?"},{"hint":"Perplexity is 2^H where H is the cross-entropy. For a test sequence, it's computed as the geometric mean of inverse probabilities: PP = (∏ 1/P(wᵢ))^(1/N). Lower perplexity indicates a better model. Unlike probability, perplexity normalizes for sequence length and provides more intuitive comparison across different test sets.","question":"What is **perplexity**, and how would you calculate it for a test corpus using your unigram model? Why is perplexity preferred over raw probability for evaluating language models?"},{"hint":"Subword tokenization breaks rare words into common subword units, reducing vocabulary size and handling out-of-vocabulary words naturally. You would apply the subword tokenizer before counting. The tradeoff is that probability assignments become more granular, and you need more sophisticated models to decode subword sequences back to words.","question":"How would you adapt this implementation to handle **subword tokenization** (like BPE or WordPiece) instead of word-level tokenization? What are the tradeoffs?"},{"hint":"Multiplying many small probabilities causes numerical underflow. Instead, compute log P(corpus) = Σ log P(wᵢ). This transforms products into sums, avoiding underflow. Use math.log or numpy.log. For a word probability P, the log probability is log(P). When the word count is zero, the log probability is negative infinity (handled specially).","question":"How would you implement **log-probability computation** instead of raw probabilities? Why is this essential for practical language modeling applications?"},{"hint":"Track word counts per document rather than globally. Term Frequency (TF) is the within-document probability, while Inverse Document Frequency (IDF) measures rarity across documents: log(total_docs / docs_containing_word). TF-IDF combines these: TF × IDF. This requires maintaining a document-term matrix or dictionary of document-specific counters.","question":"How would you modify this approach to compute **document-specific word probabilities** in a multi-document corpus, enabling TF-IDF style analysis?"},{"hint":"For streaming/online computation, maintain running counts and totals, updating probabilities incrementally. For memory-constrained scenarios, use probabilistic data structures like Count-Min Sketch for approximate frequency estimates. Space-saving algorithms can track approximate top-k frequent words. These trade exact accuracy for bounded memory usage.","question":"How would you scale this computation to handle **streaming data** where the corpus continuously grows? What approximate counting algorithms could help?"}],"realWorldApplications":[{"title":"Search Engine Query Understanding and Ranking","industry":"Search Technology & Information Retrieval","description":"Search engines like Google, Bing, and Elasticsearch use word frequency statistics to understand query intent and rank documents. **Unigram models form the foundation of TF-IDF** (Term Frequency-Inverse Document Frequency), which powers relevance scoring in information retrieval systems. When you search for a term, the engine computes how frequently that term appears in each candidate document (term frequency) relative to how rare it is across all documents. High-frequency common words like 'the' get downweighted, while rare informative words receive higher importance. Modern search engines process billions of queries daily, each requiring rapid probability computations across massive document indices."},{"title":"Spam and Content Filtering Systems","industry":"Email Security & Content Moderation","description":"Email providers and social media platforms deploy **Naive Bayes classifiers** that rely fundamentally on word probability estimation. To classify whether an email is spam, the system computes the probability of each word appearing given the class (spam vs. legitimate). These conditional probabilities are learned from labeled corpora of spam and non-spam messages. Gmail processes over 300 billion emails annually, filtering approximately 15 billion spam messages daily. 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