Probability with exponentials

$a^{b} < 0.9$ $a < 0.9^{\frac{1}{b}}$ Using geometric probability we can graph an $a$ versus $b$ graph on the interval $[0,1]$ , noting that $a$ is our independent variable and $b$ is our dependent variable. Then we find the area under the curve of this inequality. Due to the variables being switched in the equation in relation to our graph, we must take one minus that area. $\int_{0}^{1} 0.9^{\frac{1}{x}} = 0.7129$ These types of integrals are non-elementary and can only be solved by numerical methods. Thus, the answer is $1-0.7129 = \boxed{0.287}$