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Let $n$ and $1\leq k \leq n$ be natural numbers. Prove the inequality $$\sum_{i=k}^n \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i} \leq 1 - \frac{1}{e} $$

Equivalently, if $X\sim$ Bin($n$,$\frac{k}{n+1}$), prove that $\mathbb{P}[X\geq k] \leq 1 - \frac{1}{e}$.

My attempt: It may be helpful to show that the LHS tends to $1-\frac{1}{e}$ as $n \to \infty$ (already did that) and that the LHS is an increasing function on $n$ (have not done that).

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  • $\begingroup$ I don't think @gimusi's argument is correct. $\endgroup$ – DesmondMiles Mar 30 '18 at 17:40
  • $\begingroup$ did you verify after my new comments? $\endgroup$ – gimusi Apr 1 '18 at 8:47
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HINT

Note that

$$\sum_{i=k}^n \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i} =\sum_{i=0}^n \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i} -\sum_{i=0}^{k-1} \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i}=1^n-\sum_{i=0}^{k-1} \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i}$$

and show that

$$\sum_{i=0}^{k-1} \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i}\stackrel{k=1}\ge \sum_{i=0}^0 \binom{n}{i}\bigg(\frac{1}{n+1}\bigg)^i\bigg(1-\frac{1}{n+1}\bigg)^{n-i}=\bigg(1-\frac{1}{n+1}\bigg)^{n}\ge\frac 1e$$

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  • $\begingroup$ Sorry, but how did you get from $sum_{i=0}^1$ (things) to n/(n+1)*(1-1/(n+1))^{n-1}; i.e. how did k dissapear magically and became 1? $\endgroup$ – DesmondMiles Mar 30 '18 at 14:05
  • $\begingroup$ @DesmondMiles sorry there is a typo k ia assumed $=1$, I check and fix this point $\endgroup$ – gimusi Mar 30 '18 at 14:06
  • $\begingroup$ @DesmondMiles I've made also a stupid mistake with the sum index. Now it should be ok and also simpler! $\endgroup$ – gimusi Mar 30 '18 at 14:12
  • $\begingroup$ Thank you but how exactly do we show the inequality with (k=1) written above it? Apologies if I am not seeing something stupid. The gap in my attempt is quite similar. $\endgroup$ – DesmondMiles Mar 30 '18 at 14:21
  • $\begingroup$ @DesmondMiles Because it is a sum of positive terms thus the minimum is attained fro k=1, and the quantity obtained is bounded below by $1/e$. $\endgroup$ – gimusi Mar 30 '18 at 14:24

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