We have a data set with $6$ records as $2-$dimensional points, p$1$ to p$6.$ These points$$ locations in a $2-$dimensional grid and their corresponding distance matrix are given below.

Using this data we want to come up with a $3-$means clustering and, doing so$,$ we would like

to avoid the initial centroids problem. For this purpose, we choose to implement the K$-$

means$++$ algorithm. Suppose the first two centroids are p$5$ and p$1,$ respectively. Derive the

probability distribution that will be used to randomly select the third centroid $($i$.$e$.,$ calculate the probabilities of each point to be picked as the third centroid$).$ Which point is most likely to be picked? Show all your calculations. $(3\mathrm{.}5$pts$)$