What is the probability of hitting a one-fifth-area region exactly twice in three dart throws?

Instruction: Assume three independent throws with landing points uniform over the target area and no misses of the target. A specified region covers one-fifth of the area. Calculate the probability of exactly two hits in that region.

Context: This question tests the candidate's understanding of binomial probability distributions and their application to real-life scenarios.

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Official answer

Assume three independent throws, all landing on the target, with positions uniform over its area. A region occupying one-fifth of the area has hit probability 1/5; missing that region while still hitting the target has probability 4/5.

Exactly two hits in the region can occur in three orders. Therefore P(exactly two) = C(3,2)(1/5)^2(4/5) = 12/125 = 9.6%.

The 4/5 outcome is not a miss of the entire target. Also, “uniform over area” does not mean differently sized regions have equal probabilities. At least two region hits would include three hits and give 13/125.

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