Let $C$ be the unit circle $x^{2}+y^{2}=1 .$ A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diagonal $p q .$ What is the probability that no point of $R$ lies outside of $C ?$
1. **Define the points \( p \) and \( q \):**
   - Let \( p = (\cos \theta, \sin \theta) \) where \( \theta \) is uniformly distributed over \([0, 2\pi)\).
   - Let \( q = (r \cos \phi, r \sin \phi) \) where \( r \) is uniformly distributed over \([0, 1]\) and \( \phi \) is uniformly distributed over \([0, 2\pi)\).

2. **Determine the coordinates of the vertices of the rectangle \( R \):**
   - The rectangle \( R \) has sides parallel to the \( x \)- and \( y \)-axes with diagonal \( pq \).
   - The vertices of \( R \) are:
     \[
     (\cos \theta, \sin \theta), (r \cos \phi, \sin \theta), (\cos \theta, r \sin \phi), (r \cos \phi, r \sin \phi)
     \]

3. **Condition for \( R \) to lie inside \( C \):**
   - For \( R \) to lie entirely within the unit circle, all vertices must satisfy the equation \( x^2 + y^2 \leq 1 \).
   - The vertices are:
     \[
     (\cos \theta, \sin \theta), (r \cos \phi, \sin \theta), (\cos \theta, r \sin \phi), (r \cos \phi, r \sin \phi)
     \]
   - We need to check the conditions:
     \[
     (r \cos \phi)^2 + (\sin \theta)^2 \leq 1
     \]
     \[
     (\cos \theta)^2 + (r \sin \phi)^2 \leq 1
     \]
     \[
     (r \cos \phi)^2 + (r \sin \phi)^2 \leq 1
     \]

4. **Simplify the conditions:**
   - The third condition simplifies to:
     \[
     r^2 (\cos^2 \phi + \sin^2 \phi) \leq 1 \implies r^2 \leq 1 \implies r \leq 1
     \]
     This condition is always satisfied since \( r \) is uniformly distributed over \([0, 1]\).

5. **Evaluate the remaining conditions:**
   - The first condition:
     \[
     r^2 \cos^2 \phi + \sin^2 \theta \leq 1
     \]
     Since \( \sin^2 \theta \leq 1 \), this condition is always satisfied for \( r \leq 1 \).

   - The second condition:
     \[
     \cos^2 \theta + r^2 \sin^2 \phi \leq 1
     \]
     Since \( \cos^2 \theta \leq 1 \), this condition is always satisfied for \( r \leq 1 \).

6. **Conclusion:**
   - Since all conditions are satisfied for \( r \leq 1 \), the probability that no point of \( R \) lies outside of \( C \) is 1.

The final answer is \(\boxed{1}\).