{\bf Abstract:} For a set A of n applicants and a set I of m items, we consider the problem of matching applicants to items, 
where each applicant x in A provides its *preference list* defined on items.
For any matchings M and M' of the matching problem, we say that an applicant x prefers M over M' if x prefers M(x) over M'(x).
For the matching problem,
we say that M is more popular than M'$
if the number of applicants preferring
M over M' is larger than the number of applicants preferring M' over M, and define M to be a *popular matching* if there are no other matchings that are more popular than M.
We consider the extended situation,
where A is partitioned into A1, A2, ..., Ak and each Ai is assigned a weight w_i such that w_1 > w_2 > ... > w_k > 0, and polularity is calculated by using these weights.
Mahdian showed that if m > 1.42 n,
then a random instance of the matching problem has a popular matching with high probability; but nothing is known for the k-weighted matching problems.
In this paper, we analyze the k-weighted matching problems, and we show that for any beta such that m = beta n, (lower bound) if beta/n^{1/3} = o(1), then a random instance of the 2-weighted matching problems does not have a 2-weighted popular matching with probability 1 - o(1); (upper bound) if n^{1/3}/\beta = o(1), then a random instance of the 2-weighted matching problems has a 2-weighted popular matching with probability 1 - o(1).