The probability that a batsman hits a ball bowled from a fixed point is 1/4. Three such balls are bowled simultaneously and independently from that point towards the batsman. What is the probability that the batsman hits at least one of these balls?

Introduction / Context:
This problem deals with independent Bernoulli trials, where each ball has a fixed probability of being hit. Three balls are bowled simultaneously and independently, and we are interested in the probability that the batsman hits at least one of them.

Given Data / Assumptions:

• Probability that the batsman hits a single ball, p = 1/4.
• Probability that the batsman misses a single ball, q = 1 - p = 3/4.
• Three balls are bowled independently.
• We require the probability of at least one hit among the three balls.

Concept / Approach:
The probability of at least one success in multiple independent trials is usually easier to compute via the complement. The complement of at least one hit is no hit at all, meaning the batsman misses all three balls. Once we find the probability of missing all three, we subtract that from 1 to get the required probability.

Step-by-Step Solution:
Probability of hitting one ball = p = 1/4.Probability of missing one ball = q = 3/4.Events for different balls are independent.Probability of missing all three balls = q^3 = (3/4)^3.Compute (3/4)^3 = 27 / 64.Probability of at least one hit = 1 - probability of zero hits.So required probability = 1 - 27/64 = (64/64) - (27/64) = 37/64.

Verification / Alternative check:
We could also compute the probability by summing the probabilities of exactly one hit, exactly two hits and exactly three hits using the binomial distribution. However, this is more time consuming than the complement method. If we did this carefully, the sum would again simplify to 37/64, confirming the result.

Why Other Options Are Wrong:
The value 27/56 does not match the binomial pattern for three trials with probability 1/4, and it has an incorrect denominator. The fraction 11/13 is unrelated to the binomial computation, and 9/8 is greater than 1, so it cannot be a valid probability. Only 37/64 lies between 0 and 1 and matches the correct complement calculation.

Common Pitfalls:
Many learners mistakenly use 3 * (1/4) to estimate the probability of at least one hit, which is incorrect because the events overlap and are not mutually exclusive. Others forget to cube 3/4 when computing the probability of missing all three balls, using (3/4)^2 by mistake. Remember that for independent identical trials, the complement method with q^n is reliable and efficient.

Final Answer:
The probability that the batsman hits at least one of the three balls is 37/64.