If $a$ and $b$ are each randomly and independently chosen in the interval $[-1, 1]$, what is the probability that $|a|+|b|<1$?
1. **Define the problem geometrically**: We need to find the probability that $|a| + |b| < 1$ where $a$ and $b$ are chosen independently from the interval $[-1, 1]$. This can be visualized as a geometric problem in the coordinate plane.

2. **Identify the region of interest**: The region where $a$ and $b$ lie is a square with vertices at $(1,1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$. This square has an area of $4$ (since each side is $2$ units long).

3. **Describe the inequality $|a| + |b| < 1$**: The inequality $|a| + |b| < 1$ describes a diamond (or rhombus) centered at the origin with vertices at $(1,0)$, $(0,1)$, $(-1,0)$, and $(0,-1)$. 

4. **Calculate the area of the diamond**: The diamond can be divided into four right triangles, each with legs of length $1$. The area of one such triangle is $\frac{1}{2} \times 1 \times 1 = \frac{1}{2}$. Therefore, the total area of the diamond is $4 \times \frac{1}{2} = 2$.

5. **Calculate the probability**: The probability is the ratio of the area of the diamond to the area of the square. Thus, the probability is:
   \[
   \frac{\text{Area of the diamond}}{\text{Area of the square}} = \frac{2}{4} = \frac{1}{2}
   \]

The final answer is $\boxed{\frac{1}{2}}$.