In the history of Major League Baseball, what is the average score of
a game. i.e: the average number of runs scored by the winning team
and the average number of runs scored by the losing team?

According to The Baseball Archive at http://baseball1.com, through
2001 season, there has been 175665 major league games played. 
1,606,056 total runs were scored during those games.

Unfortunately, there is no breakdown as to what the total scores were
for winning/losing cause.

That translates into about 9.14 runs per game.  So the closest guess
would be that the winning teams score >5.07 runs per game and losing
teams score <4.07 runs per game on average.

peeping-ga,

thanks for the research that you have done in an attempt to arrive at
an answer to my question. However, I believe it is quite a leap to
assume that the average baseball game results in a one run victory.
One run is certainly the minimum, and the maximum is theoretically
infinite. With that in mind, I can't see how the average could be 1.

justdave_at_comcast-ga

Like I said, it's the closest guess.  Mind you, there have been tie
games in major league baseball.. not enough to make much difference in
the assertion that the run differential must be greater than or equal
to 1.