Binomial Distribution

An "at least once" question in repeated independent trials is most efficiently solved by the complement, since P(at least one hit) = 1 − P(no hits), avoiding summation over many cases. This complement strategy within the binomial model is a routine JEE Advanced shortcut. Each shot misses with probability 1 − 1/4 = 3/4, and the four shots are independent, so the probability of missing all four is (3/4)⁴ = 81/256. Therefore the probability of at least one hit is 1 − 81/256 = (256 − 81)/256 = 175/256. Option 81/256 reports the no-hit probability rather than its complement. Option 1/4 gives only a single-shot hit chance. Option 27/64 = (3/4)³ uses three shots instead of four. The reasoning combines independence, which lets the four miss probabilities multiply directly, with the complement rule for the at-least-one event; computing it the long way would mean adding the binomial probabilities of exactly one, two, three, and four hits, a far more laborious route that gives the same 175/256. This is precisely why the complement is the standard shortcut whenever the phrase at least one appears. Plausibility check: 175/256 ≈ 0.684 is a valid probability comfortably greater than the single-shot 1/4, which is expected since four independent chances substantially raise the cumulative likelihood of at least one success.