FindInstance skips solutions

Question

I have an equation to compute the bit-error rate of a binary symmetric channel: $$P_{err}(n,p) = \sum_{k = \frac{n-1}{2} + 1}^n\binom n k p^k(1-p)^{n-k}$$

I then want to find the smallest $n$-repetition code assuming that I can tolerate $P_{err}\leq\frac{1}{8}$. I tried using:

Subscript[P, err][n_, p_] = 
  Sum[Binomial[n, k]*p^k*(1 - p)^(n - k), {k, (n - 1)/2 + 1, n}];

FindInstance[Subscript[P, err][n, 1/4] < 1/8, n, Integers, 5]

But I'm getting {{n -> 555}, {n -> 1165}, {n -> 887}, {n -> 831}, {n -> 1119}} as the answer. However,

Table[{n, Subscript[P, err][n, 1/4] < 1/8}, {n, 1, 10, 2}]

Gives me {{1, False}, {3, False}, {5, True}, {7, True}, {9, True}}, showing that 5 is the first solution.

Why is this happening and how can I fix this? I'm assuming this must be something to do with the way FindInstance reacts to values of n that are even - any way to limit the search space to only odd numbers?

Answer 1

Restrict the range to find small solutions:

n /. FindInstance[Subscript[P, err][n , 1/4] < 1/8 && 0 < n < 10, n, 
  Integers, 50]

{5, 6, 7, 8, 9}

Find only odd:

(2 n + 1) /. 
 FindInstance[Subscript[P, err][2 n + 1, 1/4] < 1/8 && 0 < n < 10, n,
   Integers, 50] // Sort

{5, 7, 9, 11, 13, 15, 17, 19}