For $0<\theta<1, \theta'=(1-\epsilon)\theta, \epsilon< \theta, k\in\mathbb{N}$, the problem is to tightly upper bound the following binomial summation:
$$\sum_{i=\lceil \theta k \rceil}^k {k \choose i} (\theta')^i(1- \theta')^{(k-i)}$$.
For $0<\theta<1, \theta'=(1-\epsilon)\theta, \epsilon< \theta, k\in\mathbb{N}$, the problem is to tightly upper bound the following binomial summation:
$$\sum_{i=\lceil \theta k \rceil}^k {k \choose i} (\theta')^i(1- \theta')^{(k-i)}$$.