Given the following joint density function: $$f(x,y)=2(x+y) \\0 \le x \le y \le 1$$ Determine the probability that Y is greater than 1/2 given that X=0.1.
So, I start from the formula below: $$P(Y \gt \frac{1}{2}|X=0.1)$$
And suddenly, I realized that X is a continuous random variable. So, is P(X=0.1) = 0
Based on the solution, P(X=0.1) is not 0. Can anyone help me to understand why P(X=0.1) is not zero here?
You should use that which you have been provided: the joint probability density function.
$$\begin{align}\mathsf P(Y>0.5\mid X=0.1)&=\int_{y>0.5} f_{\small Y\mid X}(y\mid 0.1)\,\mathrm d y\\[1ex]&= \int_{y>0.5}\dfrac{f_{\small X,Y}(0.1,y)}{f_{\small X}(0.1)}\,\mathrm d y\\[1ex]&=\dfrac{\int_{0.5}^1 2(0.1+y)\,\mathrm d y}{\int_{0.1}^1 2(0.1+y)\,\mathrm d y}\end{align}$$