The man who came down with Ebola in Dallas came from Liberia to Brussels and then flew to Dulles on United Airways flight 951 on Sept 20. My daughter flew from Brussels to Dulles on flight 951 the next day, Sept 21. Is there a chance that the same exact plane was used for this same flight number and that my daughter could have contacted the same surfaces that the Ebola victim touched?
Sometimes, sometimes not. Airlines schedule their aircraft according to complex algorithms, and there's no attempt to make them match flight numbers. They may find it efficient to use the same aircraft to run the same flight day after day, or perhaps a more complicated schedule is more efficient.
For a route like Brussels-Washington, it would be unlikely to use the same aircraft every day. That route has a flying time of roughly 8 hours. If it was to be back in Brussels the next day to operate the next day's Flight 951, there are a few possibilities, none of which would be economically appealing to the airline:
Wait 8 hours in Washington and fly back to Brussels. Airlines don't like to let planes sit on the ground for such extended periods because they don't earn any money when not flying.
Fly back to Brussels immediately and wait 8 hours for the next day's departure. Same problem as before.
Fly to some third city XXX which is about 8 hours from both Brussels and Washington. City XXX would need to be outside the US, and United, like most airlines, operates almost all its international routes to and from its US hubs. So XXX-Brussels would be an unlikely route for them to want to operate.
Fly to some city YYY which is 4 hours from Washington, then back to Washington, then on to Brussels. Airlines rarely use aircraft as large, long-range, and expensive as the Boeing 777 (which is what UA uses on Brussels-Washington) on routes as short as 4 hours; it is more economical to use smaller, shorter-range planes.
So UA would likely prefer to send that aircraft somewhere else from Washington, and it would probably not be in Brussels to operate the next day's Flight 951.
In this specific case, we can actually get a definite answer. According to FlightRadar24, UA951 was operated by aircraft registration number N771UA on 20 Sep 2014, but by N781UA on 21 Sep 2014. So indeed they were two different aircraft. N771UA didn't operate UA951 again until 30 Sep.
As mentioned in comments, the information we have is that the Ebola patient had no symptoms when he flew and thus was not contagious, so there should be no risk even for someone who did fly on the same plane, or even the same flight.