Of all the possible legal chess games, how many end in white win, draw and black win (assuming 50 move rule)? If both sides make moves completely randomly but legally, how many different games are there? Since the 50 move rule applies, this number should be finite (but HUGE).

For example, there are 20 possible first moves for white to start with. If we include the first move for black, there will be 400 different positions.

Among all these different games, how many of them will be win for white? Black? Draw?

I guess very few (in term of percentage) game will end with a draw. And the winning percentage will be roughly the same for white and for black.
The number of games is huge but finite, and estimates have been made based on a number of assumptions. But that question has been asked before already, so I won't go into the details here. A short answer given on Wikipedia is at least 10123, based on an average branching factor (moves per position) of 35 and an average game length of 80; after only 10 plies (5 full moves) the exact count is 69,352,859,712,417. See Is the number of possible chess games infinite? for more answers. 

The probable outcomes of a game with random moves can be estimated "experimentally" by running random simulations. I wrote a program to do just that, had it play 1000 games, and these were my results:

Outcome                        Count  Avg. #moves
-----------------------------  -----  -----------
Draw by insufficient material    500          179
Draw by fifty-move               157          208
Draw by threefold repetition     147          164
Black wins by checkmate           75           87
White wins by checkmate           72           78
Draw by stalemate                 49          138


I think that the most common result is a draw because random moves are rather incompetent at mating, which requires the interaction of at least three pieces (counting both colors, including kings), in contrast with capturing, which is much easier to do by chance, only requiring the interaction between two pieces. After enough captures occur, you end up with insufficient material, which is the most likely outcome, occurring 50% of the time.

Alternatively, if you don't luck into a capture for long enough, you are also somewhat likely to bump into the fifty-move limit or threefold repetition. (My program assumes that these draws are claimed as soon as possible, even though they are not automatic per the rules, unlike 75-move or fivefold repetition.)

I don't expect a color advantage in random games. My sample had a few more wins with Black than with White, but I think that's within statistical noise.

For anyone who is interested, here's my program, written in Python and using the python-chess module:

import chess
import collections
import random

terminations = ['is_stalemate', 'is_insufficient_material', 'is_checkmate',
    'can_claim_fifty_moves', 'can_claim_threefold_repetition']

def get_termination(board):
    for termination in terminations:
        method = getattr(board, termination)
        if method():
            return termination

def game():
    board = chess.Board()
    for i in range(10000):
        moves = list(board.legal_moves)
        move = random.choice(moves)
        board.push(move)
        result = board.result(claim_draw=True)
        if result != '*':
            termination = get_termination(board)
            print(result, board.fullmove_number, termination)
            return result, termination

stats = collections.Counter()
for i in range(1000):
    result, termination = game()
    if termination == 'is_checkmate':
        stats[result] += 1
    else:
        stats[termination] += 1

print(stats)