Given information:
Y is distributed uniformly in the open interval (1,6).

To find:
The probability that the polynomial 3x^2 - 6xY + 3Y - 6 has only real roots.

Explanation:
To determine the probability that the given polynomial has only real roots, we need to analyze the discriminant of the quadratic equation formed by the polynomial.

Discriminant:
The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the formula: Δ = b^2 - 4ac.

Quadratic equation:
The given polynomial 3x^2 - 6xY + 3Y - 6 can be rewritten as 3x^2 - (6Y)x + (3Y - 6) = 0.

Comparing this with the standard form of a quadratic equation, we have:
a = 3, b = -6Y, and c = 3Y - 6.

Therefore, the discriminant of this quadratic equation is:
Δ = (-6Y)^2 - 4(3)(3Y - 6)
= 36Y^2 - 36(3Y - 6)
= 36Y^2 - 108Y + 216

Real roots:
For a quadratic equation to have real roots, the discriminant Δ must be greater than or equal to zero.

Therefore, Δ ≥ 0.

Solving for Δ:
36Y^2 - 108Y + 216 ≥ 0

Simplifying the inequality:
Y^2 - 3Y + 6 ≥ 0

Using the quadratic formula:
Y = (3 ± √((-3)^2 - 4(1)(6))) / 2(1)
Y = (3 ± √(9 - 24)) / 2
Y = (3 ± √(-15)) / 2

Real roots:
Since the square root of a negative number results in imaginary roots, the quadratic equation has real roots if the discriminant is non-negative.

To find the probability that the polynomial has real roots, we need to find the range of Y values that satisfy the inequality Y^2 - 3Y + 6 ≥ 0.

Solving the inequality:
Y^2 - 3Y + 6 ≥ 0

Plotting the quadratic equation on a graph, we see that the parabola opens upwards and the range of Y values that satisfy the inequality is when the graph is above the x-axis.

The graph intersects the x-axis at Y = (3 ± √(-15)) / 2, which means the parabola does not intersect the x-axis for any real Y values. Therefore, there are no real roots for the polynomial.

Probability:
Since the polynomial has no real roots, the probability is 0.

Therefore, the probability that the polynomial 3x^2 - 6xY + 3Y - 6 has only real roots is 0.