History demonstrates that it is typical for a Democratic President to select a Republican for his cabinet, with the Defense Department the usual position awarded to a member of the opposition party.
This was true with Robert McNamara, who was Secretary of Defense under John F. Kennedy and Lyndon B. Johnson.
The same scenario occurred with William Cohen serving under Bill Clinton in his second term.
And under Barack Obama, we saw carryover Robert Gates and later Chuck Hagel head the Pentagon.
If we go back to before the Defense Department was created in 1947, it was known as the War Department, and Republican Henry Stimson served as Secretary of War for Franklin D. Roosevelt, and Republican Robert Lovett served for two years in that position under Harry Truman.
Only Jimmy Carter among Democratic Presidents never had a Republican serve as Secretary of Defense, but he did have James Schlesinger as the first Secretary of Energy.
So it seems likely that Joe Biden will pick a Republican for the Cabinet, and the most likely choice would be former Congressman and Ohio Governor John Kasich, who will deliver a speech of endorsement at the Democratic National Convention on Monday night.