This expression for an educated guess or a close estimate comes from the American ball fields.
In the old days, baseball games were only played during daylight hours. News reporters covering the games wanted to know the attendance so they could report it in the evening paper. Managers would never want to reveal a low figure, so they guessed - giving a broad or ballpark figure.
But where did "ballpark" come from? Baseball stadiums were called ballparks beginning in the early 1900s. Before that they were ball grounds and ball fields. "Baseball stadium" may be popular usage today, but "ballpark" prevails in expressions like "in the ballpark," meaning well within reach.
Which came first - the cart or the athletic trainer?
The little village of Kocs in Hungary is responsible for the origin of "coach," meaning closed cart or traveling carriage. A local carriagemaker came up with an idea for a large, comfortable vehicle and named it a "Kocs cart" or kocsi. Copied all over Europe in the early centuries, its name changed to coche in French and then was anglicized to "coach" by 1550.
The nickname for a private tutor arose in English universities centuries later - to "carry" students though exams, maybe? The sense of an "athletic tutor" or coach followed close behind.
But what about the stagecoach? The original meaning of "stage" was a "stop," as in a stopping place on a journey. "Stage" also became attached to the theater because the traveling companies of the 15th century journeyed by cart and put on performances at their stopping places, or stages. Soon "stage" meant not only the stop, but the leg of a journey, and then finally the vehicle in which the journey was made.
SOURCES: 'Why You Say It,' by Webb Garrison; 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson; 'The Morris Dictionary of Word and Phrase Origins,' by W. and M. Morris; 'The Barnhart Dictionary of Etymology,' by Robert K. Barnhart; 'Dictionary of Word Origins,' by Joseph Shipley.