**Core Concept**
The concept of conditional probability is being tested here, specifically the calculation of the probability of an event (picking up a person requiring surgery) given another event (picking a girl or a boy). This requires understanding of the Bayes' theorem and its application in real-world scenarios.

**Why the Correct Answer is Right**
To solve this problem, we need to calculate the probability of picking a person requiring surgery given that they are a girl or a boy. We know that 10 out of 20 girls (50%) and 20 out of 30 boys (66.7%) required surgery. Using the formula for conditional probability, we can calculate the probability as follows: (Number of girls requiring surgery / Total number of girls) / (Number of girls requiring surgery / Total number of girls) + (Number of boys requiring surgery / Total number of boys) = (10/20) / (10/20 + 20/30) = 0.5 / (0.5 + 0.6667) = 0.5 / 1.1667 = 0.4286. This is approximately equal to 0.43 or 43%.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because it does not take into account the conditional probability of picking a person requiring surgery given that they are a girl or a boy. It simply calculates the probability of picking a girl or a boy, which is not relevant to the question.

**Option B:** This option is incorrect because it assumes that the probability of picking a person requiring surgery is the same for girls and boys, which is not the case given the different percentages of girls and boys requiring surgery.

**Option C:** This option is incorrect because it calculates the probability of picking a girl or a boy requiring surgery, but does not take into account the total number of girls and boys in the hospital.

**Option D:** This option is incorrect because it does not provide a numerical value and is therefore not a valid answer.

**Clinical Pearl / High-Yield Fact**
When dealing with conditional probability problems, make sure to identify the relevant events and their probabilities, and use the Bayes' theorem to calculate the desired probability.

**Correct Answer:** C.