lean_workbook_plus_19449

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $P(n)$ be the probability that a series of $n$ tosses have no consecutive heads. Then If we threw a tail, we need the rest of the sequence to have no consecutive heads. However, If we threw a head, we must throw a tail next, and a sequence that has no consecutive heads. As the result, we get $P(n) = \frac{1}{2}(P(n-1)) + \frac{1}{2}(\frac{1}{2}P(n-2))$. Obviously, $P(1) = 1$, and $P(2) = \frac{3}{4}$ (I'm not brave enough to assume $P(0) = 1$ ...), now work our way back up we get $P(10) = \frac{281}{512}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_19449  (p : ℕ → ℚ)
  (h₀ : p 1 = 1)
  (h₁ : p 2 = 3 / 4)
  (h₂ : ∀ n, p (n + 2) = 1 / 2 * (p n) + 1 / 2 * (1 / 2 * (p (n - 1)))) :
  p 10 = 281 / 512   :=  by sorry

There are two separate errors: formal recurrence indexing and the source’s final numerical value.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use P(n+2)=P(n+1)/2+P(n)/4 with P(1)=1,P(2)=3/4, and replace the source’s incorrect numerical answer by P(10)=9/64.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.