# Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$.

Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I get at most $T$ heads is

$$\left(1-\dfrac1P\right)^{PR}+\binom{PR}{1}\left(1-\dfrac1P\right)^{PR-1}\left(\dfrac1P\right)+\ldots+\binom{PR}{T}\left(1-\dfrac1P\right)^{PR-T}\left(\dfrac1P\right)^T$$

What upper bound can we find for this?

In particular, it would be nice to get an upper bound that is independent of $P$, something like $$\binom{aT+R}{R}e^{bR+cT}$$ for some constants $a,b,c$.

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cs.anu.edu.au/~bdm/papers/littlewood2.pdf might be useful. –  Stephen Montgomery-Smith Nov 20 '13 at 20:49
@StephenMontgomery-Smith I don't see how anything there might be useful. Could you be more specific? –  Kunal Nov 20 '13 at 21:03
No, it was just a passing comment. –  Stephen Montgomery-Smith Nov 20 '13 at 21:37