From a group of 5 boys and 4 girls (9 people in total), a group of 5 persons is formed at random. What is the probability that the selected group contains at least two girls?

Introduction / Context:
This question combines combinations and probability in a selection problem involving boys and girls. You are asked to find the probability that a randomly chosen group of five people contains at least two girls. Such problems are standard in aptitude tests and train you to think in terms of counting favorable combinations and using complements when convenient.

Given Data / Assumptions:

• Number of boys = 5.
• Number of girls = 4.
• Total people = 9.
• We select 5 people at random from these 9.
• We want groups that contain at least two girls.

Concept / Approach:
The total number of ways to select any 5 people from 9 is 9C5. Instead of counting all cases with at least two girls directly, it is simpler to use the complement rule: At least two girls = Total groups - (groups with zero girls + groups with exactly one girl). We compute each of these counts using combinations and then convert the result into a probability.

Step-by-Step Solution:
Step 1: Total number of groups of 5 from 9 people = 9C5. 9C5 = 9 * 8 * 7 * 6 / (4 * 3 * 2 * 1) = 126. Step 2: Count groups with zero girls (all boys). To have zero girls, we must choose all 5 people from the 5 boys: this is 5C5 = 1 group. Step 3: Count groups with exactly one girl. Choose 1 girl from 4: 4C1 = 4. Choose the remaining 4 members from the 5 boys: 5C4 = 5. So groups with exactly one girl = 4 * 5 = 20. Step 4: Total groups with fewer than two girls = 1 + 20 = 21. Step 5: Groups with at least two girls = total groups minus these = 126 - 21 = 105. Step 6: Probability of at least two girls = 105 / 126. Step 7: Simplify 105 / 126 by dividing numerator and denominator by 21 to get 5 / 6.

Verification / Alternative check:
You can cross check by explicitly listing the possible numbers of girls in the group: 2, 3, or 4. Then compute:

• Exactly two girls: 4C2 * 5C3.
• Exactly three girls: 4C3 * 5C2.
• Exactly four girls: 4C4 * 5C1.

Adding these three values and dividing by 9C5 also gives 5/6. This second method confirms that the complementary counting approach is consistent and accurate.

Why Other Options Are Wrong:
Option 2/3, option 3/5, and option 11/18 are all less than the true probability and arise from various partial counts or missing cases. Option 7/9 is larger than 5/6 and would imply that almost all groups meet the condition, which is not correct. None of these values agree with the complete combination counting or the complement approach, while 5/6 does.

Common Pitfalls:
A frequent mistake is to misinterpret at least two girls as exactly two girls, ignoring cases with three or four girls. Another common error is to forget to subtract both zero girls and one girl cases when using the complement method. Also, when working quickly, students sometimes calculate 9C5 incorrectly, which then affects every subsequent ratio. Careful use of combinations and explicit consideration of all relevant cases can prevent these errors.

Final Answer:
The probability that a randomly chosen group of five people contains at least two girls is 5/6.