Ten equally spaced point are lying on a circle, if p is the probability that a chosen chord is diameter of the circle, then 18p+2 is equal to ______.

Answer

To solve the problem, we need to find the probability \( p \) that a randomly chosen chord from 10 equally spaced points on a circle is a diameter. Then, we will calculate \( 18p + 2 \).

### Step-by-Step Solution:

1. **Identify the Points on the Circle**:
We have 10 equally spaced points on the circumference of a circle. Let's label these points as \( P_1, P_2, P_3, P_4, P_5, P_6, P_7, P_8, P_9, P_{10} \).

2. **Determine the Total Number of Chords**:
A chord can be formed by selecting any two points from these 10 points. The number of ways to choose 2 points from 10 is given by the combination formula:
\[ \text{Total Chords} = \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45 \]

3. **Identify the Diameter Chords**:
A chord is a diameter if it connects two points that are directly opposite each other on the circle. Given 10 points, each point has exactly one point that is directly opposite it. Thus, the pairs of points that form diameters are:
- \( (P_1, P_6) \)
- \( (P_2, P_7) \)
- \( (P_3, P_8) \)
- \( (P_4, P_9) \)
- \( (P_5, P_{10}) \)

Therefore, there are 5 diameter chords.

4. **Calculate the Probability \( p \)**:
The probability \( p \) that a randomly chosen chord is a diameter is given by the ratio of the number of diameter chords to the total number of chords:
\[ p = \frac{\text{Number of Diameter Chords}}{\text{Total Chords}} = \frac{5}{45} = \frac{1}{9} \]

5. **Calculate \( 18p + 2 \)**:
Now, we substitute \( p \) into the expression \( 18p + 2 \):
\[ 18p + 2 = 18 \times \frac{1}{9} + 2 = 2 + 2 = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]