Out of 17 applicants for a job (8 boys and 9 girls), two persons are to be selected at random. What is the probability that at least one of the selected persons will be a girl?

Introduction / Context:
This question illustrates the use of combinations and the complement rule in probability. We are selecting two persons at random from a group of applicants consisting of boys and girls, and we want the probability that there is at least one girl among those selected. Often, it is easier to find the probability of the complementary event (no girl selected) and subtract this from 1 to get the final answer.

Given Data / Assumptions:

Total applicants = 17.

Boys = 8.

Girls = 9.

Two persons are chosen at random.

We are interested in the event that at least one of the selected persons is a girl.

Concept / Approach:
Instead of counting combinations that contain at least one girl directly, we consider the complement event: both selected persons are boys. The probability of at least one girl is then 1 minus the probability that both selected are boys. This method keeps the counting simple and reliable, especially when there are many ways to include at least one girl but only a few ways to include no girls.

Step-by-Step Solution:
Step 1: Total number of ways to choose 2 persons out of 17 = 17C2.Step 2: Compute 17C2 = (17 * 16) / (2 * 1) = 136.Step 3: For the complement event (no girl selected), both selected persons must be boys.Step 4: Number of ways to choose 2 boys out of 8 = 8C2.Step 5: Compute 8C2 = (8 * 7) / (2 * 1) = 28.Step 6: Probability that both selected are boys = 28 / 136.Step 7: Simplify 28 / 136 by dividing numerator and denominator by 4 to get 7 / 34.Step 8: Probability that at least one of the selected persons is a girl = 1 - 7 / 34 = (34 - 7) / 34 = 27 / 34.

Verification / Alternative check:
We can double check by computing probabilities directly. The probability that the first selected person is a boy is 8 / 17, and if this happens, the probability that the second is also a boy is 7 / 16. Thus, P(both boys) = (8 / 17) * (7 / 16) = 56 / 272 = 7 / 34. Taking 1 minus this value again gives 27 / 34 for at least one girl, confirming the combination based method.

Why Other Options Are Wrong:
The value 19/34 or 25/34 would arise from incorrect subtraction or miscounting the all boy combinations. The fraction 20/34 simplifies to 10/17 and does not correspond to any correct counting argument. The value 7/34 is the probability of selecting two boys and hence represents the complement event, not the event with at least one girl.

Common Pitfalls:
Some learners accidentally count the cases with exactly one girl and exactly two girls separately and make arithmetic errors when adding. Others forget that the group contains more girls than boys and guess that the probability is close to 1/2. The complement approach, focusing on the simpler event of choosing two boys, is usually the safest and fastest route.

Final Answer:
The probability that at least one of the two selected persons is a girl is 27/34.