Female Teachers

Let $x$ be the total number of participants at the meeting.

Since $\frac{1}{3}$ of the participants are men, the number of men is $\frac{x}{3}$ .

Since $\frac{1}{2}$ of the participants are teachers, the number of teachers is $\frac{x}{2}$ .

Since $80\%$ of the teachers are women, the number of female teachers is $80\%$ of $\frac{x}{2} = \frac{2x}{5}$ .

The number of male teachers is the number of teachers minus the number of female teachers, which is $\frac{x}{2} - \frac{2x}{5} = \frac{x}{10}$ .

Since $\frac{1}{2}$ of the men are engineers, the number of male engineers is $\frac{1}{2} \cdot \frac{x}{3} = \frac{x}{6}$ .

The number of male doctors is the number of men minus the number of male teachers minus the number of male engineers, which is $\frac{x}{3} - \frac{x}{10} - \frac{x}{6}= \frac{x}{15}$ .

Since the number of female doctors is equal to the number of male doctors, the number of female doctors is $\frac{x}{15}$ .

The probability of being a female doctor is therefore $\frac{\frac{x}{15}}{x} = \frac{1}{15} \approx \boxed{0.0666}$ .