Question 1210173
Let's break down this problem step by step.

**1. Geometric Interpretation**

We are looking for the probability that BD < 4. This is equivalent to finding the area of the region inside triangle ABC where BD < 4, and dividing it by the total area of triangle ABC.

**2. Triangle ABC and Ray BP**

* Let P be a random point inside triangle ABC.
* Ray BP intersects AC at point D.
* We want to find the probability that BD < 4.

**3. Law of Cosines**

Let's find the angles of triangle ABC using the Law of Cosines:

* Let a = 3, b = 5, c = 7.
* $\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{25 + 49 - 9}{2(5)(7)} = \frac{65}{70} = \frac{13}{14}$
* $\cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{9 + 49 - 25}{2(3)(7)} = \frac{33}{42} = \frac{11}{14}$
* $\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{9 + 25 - 49}{2(3)(5)} = \frac{-15}{30} = -\frac{1}{2}$

Since $\cos C = -1/2$, angle C = 120 degrees.

**4. Area of Triangle ABC**

We can use Heron's formula to find the area of triangle ABC.

* Semi-perimeter: $s = (3 + 5 + 7)/2 = 15/2 = 7.5$
* Area = $\sqrt{s(s-a)(s-b)(s-c)} = \sqrt{7.5(4.5)(2.5)(0.5)} = \sqrt{42.1875} = \frac{15\sqrt{3}}{4}$

**5. Geometric Reasoning**

* Let's consider the locus of points P inside triangle ABC such that BD = 4.
* We can create a triangle ABD' where BD' = 4.
* The locus of P such that BD = 4 is a line parallel to AD'.
* The region where BD < 4 is the region inside triangle ABC that is closer to B than to D'.

**6. Ratio of Areas**

* The probability we are looking for is the ratio of the area of the region where BD < 4 to the area of the triangle ABC.
* Let's consider a point P that makes BD = 4.
* Let's determine the ratio of AD/CD.
* By the Angle Bisector Theorem, if P were to lie on an angle bisector, we could find the ratios. However, P is not necessarily on an angle bisector.
* The ratio AD/CD is dependent on the position of point P.

**7. Difficulties**

* Determining the exact shape of the region where BD < 4 is challenging without further geometric constructions and calculations.
* The problem requires a more advanced approach, potentially involving barycentric coordinates or other geometric methods.

**8. Conclusion**

Without further information or a more advanced approach, it's difficult to find the exact probability. However, we know that the probability is the ratio of the area of the region where BD < 4 to the total area of the triangle.

Unfortunately, I don't see a way to solve this problem without more advanced techniques.