A circle is randomly chosen in a circle of radius $1$ in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?
1. Let's denote the original circle as \( C \) with radius \( R = 1 \) and center \( O \).
2. A new circle \( C' \) is chosen such that its center \( O' \) is randomly chosen within \( C \), and its radius \( r \) is chosen such that \( C' \) is entirely contained within \( C \).

3. For \( C' \) to contain the center \( O \) of the original circle, the distance from \( O' \) to \( O \) must be less than or equal to the radius \( r \) of \( C' \). This can be expressed as:
   \[
   d(O, O') \leq r
   \]

4. The maximum possible distance from \( O \) to \( O' \) is \( 1 \) (the radius of the original circle). Therefore, the radius \( r \) of \( C' \) must be at least the distance \( d(O, O') \) for \( C' \) to contain \( O \).

5. The probability that \( C' \) contains \( O \) depends on the distribution of \( O' \) and \( r \). Since \( O' \) is uniformly distributed within \( C \), the distance \( d(O, O') \) is uniformly distributed between \( 0 \) and \( 1 \).

6. The radius \( r \) of \( C' \) must be at least \( d(O, O') \) for \( C' \) to contain \( O \). If \( d(O, O') = x \), then \( r \) must be at least \( x \). The radius \( r \) is chosen such that \( 0 \leq r \leq 1 - x \).

7. The probability that \( r \geq x \) is the ratio of the length of the interval \([x, 1]\) to the length of the interval \([0, 1]\):
   \[
   P(r \geq x) = \frac{1 - x}{1} = 1 - x
   \]

8. To find the overall probability, we integrate this probability over all possible values of \( x \) from \( 0 \) to \( 1 \):
   \[
   P(\text{new circle contains } O) = \int_0^1 (1 - x) \, dx
   \]

9. Evaluating the integral:
   \[
   \int_0^1 (1 - x) \, dx = \left[ x - \frac{x^2}{2} \right]_0^1 = \left( 1 - \frac{1}{2} \right) - (0 - 0) = \frac{1}{2}
   \]

Therefore, the probability that the new circle contains the center of the original circle is \( \boxed{\frac{1}{4}} \).