A light switches on and off at a fast rate, with equal time off and on, why do we see a light that appears only on rather than only off?
This is a question for a biology board.
It is because of the way the cells in our retina work. Have you heard of the "after image"? If one sees a bright enough objects and shuts ones eyes, an after image appears with eyes closed.
The same biological mechanism keeps the sense of light, rather than dark and bridges over the gap. I am guessing at the physics here, but it must be due to the chemistry of the organic molecules and how it is used to transmit to the head the information On and Off. Obviously once excited to an on state, there is a lifetime to decay to the off, and depending on the frequency from a certain point onward it bridges the gap.
Something like that happens also with the change of frames in the film of movies and other optical motions.
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The presupposition of your question is incorrect. Normal household lighting is not off half the time. household voltage is an AC voltage, provided at 60Hz (in the U.S.; 50Hz in many countires outside the U.S.). Incandescent and flouresccent lighting use both the positive and negative going portions of the wave, and therefore would appear (if viewed in slow motion) to flicker at 120Hz. But the percentage of time that the voltage is at zero (off) is much less than 1% of the time.
For incandescent lighting, the light comes from a heated filament. When current is removed, this filament cools (and gives less light) at an exponential decay rate, with a time constant on the order of 10mS. But the period of the 120 Hz power lobes is less than 10ms. That is roughly 1 time constant of the decay rate. So, you see, the filament never gets a chance to cool off and stop giving light: The light is on 100% of the time.
I am not an expert on flourescents, so I will skip the opportunity to embarrass myself, and go on to something which will make that discussion moot: LEDs.
LEDs ~DO~ turn on and off instantly. So say you had a string of 50 to 70 LEDs in series, and another identical string run in parallel but with polarity reversed. Put a current limiting resistor on the string, and (with careful engineering) you have something which DOES turn ON and stay OFF about 50% of the time. In that case you STILL would perceive it to be on all the time, since your eye cannot track the lighting changes and acts as an averaging device. You perceive the light to be ON all the time, but at half the brightness that you would perceive if they were, indeed, ON all the time.
If we vary the duty cycle, say 10% ON and 90% OFF, you would still perceive the LEDs to be ON all the time, just much dimmer. By varying the amount of time ON (the width of the "ON" pulse), we can make the string appear any brightness between 0 and 100%. This is called Pulse Width Modulation (PWM) and is a common technique for digitally varying the power to a device to mimic analog scaling. It works extremely well when the target device is the human eye.
Remember that the ON/OFF cycles are about 120 Hz, but the human eye-brain system can not track much over 20 Hz. When you see a motion picture (a movie) you are not watching motion at all, but a series of stills flashed up in rapid succession, somewhere in the neighborhood of 30 pictures a second. Do you remember all 30 pics in that second? No. You perceive that as 1 second of fluid motion. How much more so, then, will you perceive light flickering at 120 Hz to be a smooth and constant light level?