Most integrated flight displays – the so-called glass cockpit – contain multiple redundancies both on the forms of computer power, and electrical power sources to keep them operating in the event of an emergency or other electrical problems.
As an example, I obtained my multi engine add-on in a Diamond DA-42 TwinStar airplane equipped with a Garmin G1000 glass cockpit. The aircraft’s electrical system uses a primary 24 V battery which is supplies power to all the electrical buses in the aircraft. It is also charged by two 24V, 60 amp alternators, one on each engine, which supply power to their own electrical bus and the battery bus. The main electrical busses supply power to the avionics power bus and both alternators can feed this bus as well. In the event of a total electrical failure to both the battery and alternator power buses, there is an emergency battery available to power the avionics for at least 30 minutes of continued operation.
Many aircraft carry separate back up EFIS displays, each with its own emergency power supply for additional redundancy.
If you compare this with the typical systems an twin aircraft with a conventional cockpit, most of the gyroscopic instruments are powered by a pair of vacuum pumps, one on each engine. The turn and bank indicator is powered by the aircraft’s electrical system. Lose both of the vacuum pumps, and you’re on partial panel.
In comparing these two systems you would have to look at the probability of an all out failure for both systems. The new glass cockpits have been just as reliable, if not more reliable, then conventional steam gauges and far more functional in the event of a failure.