# Taking assigning an unknown variable $\theta = 0.5$ to solve an inequality

In order to solve the following inequality for $$n$$, where $$0 < \theta < 1$$, they use a trick that I do not understand

$$n \geq (\frac{1.96}{0.03})^2 \theta(1 - \theta)$$

Since $$\theta$$ is unknown we take $$\theta=0.5$$ so the inequality is true for all $$0 < \theta < 1$$. Thus

$$n \geq (\frac{1.96}{0.03})^2 (0.5)^2 = 1067.1$$

Why are they allowed to set $$\theta = 0.5$$? Why does that value of $$\theta$$ make the inequality hold for all $$0 < \theta <1$$?

• In general, drawing a little picture is a good idea. In this case, plotting $x \mapsto x(1-x)$ will give a hint. – copper.hat Jun 9 at 20:56

The value of $$\theta(1-\theta)$$ is maximal when we set $$\theta=\frac12$$. That is the reasoning justifying that step.
(Think of a sketch of the parabola $$x(1-x)$$).