Question

\(20\) girls, among whom are \(A\) and \(B\) sit down at a round table. The probability that there are \(4\) girls between \(A\) and \(B\) is

• A. \(\frac{2}{19}\)
• B. \(\frac{6}{19}\)
• C. \(\frac{13}{19}\)
• D. \(\frac{17}{19}\)

Hint:

For any relative position \(k\) in a circle of \(n\) people, if \(k<(n-1)/2\), the probability of person \(B\) being at that specific relative position from \(A\) is always \(\frac{2}{n-1}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In a circular arrangement of \(n\) items, we fix one item to remove rotational symmetry. The probability then depends on the remaining available positions for the second item.
Step 2: Key Formula or Approach:
\[ P(\text{Event}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} \]
Step 3: Detailed Explanation:
1. Let girl \(A\) be fixed at any specific seat at the round table.
2. There are \(20 - 1 = 19\) other seats remaining for girl \(B\) to occupy. So, total outcomes = \(19\).
3. For there to be exactly \(4\) girls between \(A\) and \(B\), \(B\) must be in the 5th position from \(A\).
4. This can occur in two directions:
- 5th seat to the left of \(A\).
- 5th seat to the right of \(A\).
5. Thus, there are exactly \(2\) favorable positions for girl \(B\).
6. Probability = \(\frac{2}{19}\).
Step 4: Final Answer:
The probability is \(\frac{2}{19}\).