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I think it is easier if you rewrite your inequality as \begin{align} & P( \bigcap_{i=1}^{s-1}A_i ) - P ( \bigcap_{i=1}^{s-1}A_i \cap A_s) \geq P(\bigcap_{i=1}^{s-1}A_i)(1-P(A_s)) \\ & \iff P(A_s) \geq P(A_s | \bigcap_{i=1}^{s-1}A_i). \end{align} Now let $\{X_i\}_{i=1}^n$ be Bernouli independent random variables with success probability $1/n$ and $N_{s-1}$ is the random variable which counts how many times the letters $\{1,\dots,s-1\}$ appear in a given random word. Then the LHS is exactly $$P(\sum_{i=1}^nX_i \geq p )$$ while the RHS is exactly $$ P(\sum_{i=1}^{n-N_{s-1}}X_i\geq p),$$ which is clearly smaller.