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The probability of a man hitting the target is 1/4, how many times he should shoot so that his probability of hitting the target at least once becomes 3/4.

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What is the probability that he does not hit the target at all in $n$ tries? As a function of $n$, I mean. And how is this related to the probability that he hits the target at least once on these same $n$ tries? – Dilip Sarwate Apr 27 '12 at 14:51

Hints: Let $E_n$ denote the event that the a man will shoot the target at least once after $n$ trials.

  • Find the probability of the complement of $E_n$, assuming shootings are independent.

  • Compute $\mathbb{P}(E_n)$ from $\mathbb{P}(E_n^c)$

  • Determine for which $n$ you would have $\mathbb{P}(E_n) > \frac{3}{4}$.

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