Assume you observe two colonies of ants, $C_1$ and $C_2$, in a forest. Their anthills are situated $140$ meters one from each other. There are $n_1 = 200$ ants in the colony $C_1$ and $n_2 = 200$ ants in the colony $C_2$. The position of every ant is recorded at a fixed time (using a special digital camera after ants in each colony have been marked with a fluorescent product, red and blue respectively). The random position of the $i^{\text{th}}$ ant from colony $C_1$ is given under the form $(X_{1i},Y_{1i})$. Similarly, the random position of the $j^{\text{th}}$ ant from colony $C_2$ is denoted $(X_{2j},Y_{2j})$. For all coordinates, the center of origin is taken to be the location of the anthill from colony $C_1$. Histograms of the $x$ and $y$ coordinates suggest that one can assume that $X_1$ and $Y_1$ both follows a $N(0,30)$ distribution, and scatterplots suggest they are independent, while $X_2$ and $Y_2$ both follows a $N(100,30)$ distribution and are independent. A biologist who studies these two colonies claims that ants from one colony have very little interaction with ants of the other colony.
What is the probability that the distance between two ants chosen at random, one from each colony, is less than 70 meters? That is compute, or approximate, Question.
