4. If Xiaozhang's sleep time each day is randomly and uniformly distributed between $6 \sim 9$ hours, then the probability that Xiaozhang's average sleep time over two consecutive days is no less than 7 hours is $\qquad$ .
4. $\frac{7}{9}$.

Let the sleep times of Xiao Zhang for two consecutive days be $x, y$
$$
(x, y \in [6,9]) \text{ hours. }
$$

The possible events are denoted as $(x, y)$, corresponding to the points on the sides and inside the square $A B C D$ in the coordinate plane.

Xiao Zhang's average sleep time over two consecutive days is no less than 7 hours, i.e., $x+y \geqslant 14$, and $(x, y)$ corresponds to the points on the sides and inside the pentagon $E B C D F$, as shown in Figure 2.

Therefore, the probability of the desired event is
$$
1-\frac{S_{\triangle A E F}}{S_{\text {square } A B C D}}=1-\frac{2}{9}=\frac{7}{9} \text {. }
$$