/**
 * One-sided binomial tail: the probability of getting at least k correct out
 * of n trials by guessing (p = 1/2 per trial). Computed in log space so it
 * stays finite for large n.
 */
export function binomialTailP(k: number, n: number): number {
  if (n <= 0) return 1
  if (k <= 0) return 1
  let logTerm = -n * Math.LN2 // log C(n,0) 0.5^n
  let cum = 0
  for (let i = 0; i <= n; i++) {
    if (i >= k) cum += Math.exp(logTerm)
    if (i < n) logTerm += Math.log(n - i) - Math.log(i + 1)
  }
  return Math.min(1, cum)
}

export function formatP(p: number): string {
  if (p >= 0.01) return p.toFixed(2)
  if (p >= 1e-4) return p.toFixed(4)
  return p.toExponential(1)
}