Complete the nuclear decay equation for beta-particle emission and determine the half-life from the graph.

$$\boxed{{}_{81}^{208}\text{Tl} \rightarrow {}_{82}^{208}\text{Pb} + {}_{-1}^{0}\beta}$$

b) To determine the half-life from the graph, we must first account for the background count.

The initial counter reading at $t=0$ is 80 counts per minute.
The background count is 8 counts per minute.
Initial actual activity = Counter reading - Background count

$$\text{Initial actual activity} = 80 - 8 = 72 \text{ counts per minute}$$

After one half-life, the actual activity will be half of the initial actual activity.

$$\text{Activity after one half-life} = \frac{72}{2} = 36 \text{ counts per minute}$$

This is the value we will find on the y-axis of the graph.

$$\text{Counter reading} = \text{Actual activity} + \text{Background count}$$ $$\text{Counter reading} = 36 + 8 = 44 \text{ counts per minute}$$

Locate 44 counts per minute on the y-axis.
Draw a horizontal line from 44 counts per minute to intersect the curve.
From the intersection point on the curve, draw a vertical line down to the x-axis.
Read the time value on the x-axis. This value is the half-life.
From the graph, a counter reading of 44 counts per minute corresponds to a time of approximately 3.3 minutes.

The half-life of thallium-208 is $\boxed{\text{3.3 minutes}}$.