Probability Problem Decomposition Chain

validated scientific chain v1.0.0 cc-by-sa

Block ID: b9389e95-af0a-4d6a-bd78-dda6e4db4725

Community-contributed block. PromptDNA makes no guarantee of output quality or fitness for purpose. User assumes all responsibility for use.

Template

Solve the probability problem in {probability_problem} by decomposing it carefully, since probability intuition is famously unreliable. (1) Define the sample space and the event of interest in precise terms, and check the crucial framing question: are events independent or dependent, with or without replacement, ordered or unordered? Getting this wrong at the start guarantees a wrong answer no matter how careful the arithmetic. (2) Identify whether the problem is asking for a joint, marginal, or conditional probability, and rewrite any "given that" language as an explicit conditional — conditioning that's done implicitly is done wrong. (3) Choose the tool: direct counting for finite equally-likely outcomes, the complement when "at least one" appears (computing the complement is usually far easier), conditional probability and Bayes where information updates, or a distribution where a standard model fits. (4) Execute, showing the counting or the probability multiplication, and watch for double-counting overlapping cases (use inclusion-exclusion) and for miscounting arrangements (permutations versus combinations). (5) Sanity-check: the answer must lie in [0,1], and extreme cases should give sensible limits. (6) Where intuition and calculation clash, trust the calculation but find why intuition misled — that's often where the subtlety lives.

Solve the probability problem in {probability_problem} by decomposing it carefully, since probability intuition is famously unreliable. (1) Define the sample space and the event of interest in precise terms, and check the crucial framing question: are events independent or dependent, with or without replacement, ordered or unordered? Getting this wrong at the start guarantees a wrong answer no matter how careful the arithmetic. (2) Identify whether the problem is asking for a joint, marginal, or conditional probability, and rewrite any "given that" language as an explicit conditional — conditioning that's done implicitly is done wrong. (3) Choose the tool: direct counting for finite equally-likely outcomes, the complement when "at least one" appears (computing the complement is usually far easier), conditional probability and Bayes where information updates, or a distribution where a standard model fits. (4) Execute, showing the counting or the probability multiplication, and watch for double-counting overlapping cases (use inclusion-exclusion) and for miscounting arrangements (permutations versus combinations). (5) Sanity-check: the answer must lie in [0,1], and extreme cases should give sensible limits. (6) Where intuition and calculation clash, trust the calculation but find why intuition misled — that's often where the subtlety lives.

Variables

Name	Type	Required	Trust level
probability_problem		yes

probabilitycombinatoricsconditionalchain-of-thought

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Benchmarks

Not yet self-validated against any benchmark. Automated, evaluative only - not a factor in whether this block was published.

Submitted by James P FounderMod via mcp · 2026-07-18