2. In the region $\left\{\begin{array}{l}0 \leqslant x \leqslant 2 \pi, \\ 0 \leqslant y \leqslant 3\end{array}\right.$, a point $P(a, b)$ is randomly taken, the probability that $b>\left(\sin \frac{a}{2}+\cos \frac{a}{2}\right)^{2}$ is $ـ$. $\qquad$
2. $\frac{2}{3}$.

Consider the function
$$
y=\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}=1+\sin x,
$$

and the region $\left\{\begin{array}{l}0 \leqslant x \leqslant 2 \pi, \\ 0 \leqslant y \leqslant 3\end{array}\right.$ with area $6 \pi$.
By symmetry and cutting and pasting, we know that the area of points $P(a, b)$ satisfying
$$
b>\left(\sin \frac{a}{2}+\cos \frac{a}{2}\right)^{2}
$$

is $4 \pi$, hence the probability is $\frac{2}{3}$.