I have heard so many people say that OCing decreases MTBF that I know it to be true. I thought to myself, maybe it works the other way, maybe UCing or underclocking will make my PC last forever. What do you guys think about that?

Nope

Never happen. Computer components are designed with a shelf life in mind more often then not when everything goes according to design the component will out live its usefulness. The hardware will always due what you bought it to do. Its the constant wanting it to do more that causes them to become obsolete. If all I wanted to do was play Shamus and write in basic my IBM PC Jr might still be alive today.

If something has an MTBF of 50,000 hours, does that mean it will run after I turn it on once and leave it on for 50k hours or if I constantly turn it on and off with 50k hours?

MTBF stands for 'Mean Time Before Faillure'. It is an average (mean) score, rounded down somewhat. In a worst case scenario example, it would mean that some of those components would give out just after one hour, and some others would still work after 99k hours.

It is a number that is plucked out of the air, representing an amount of time a component will be expected to work, considering *normal* use of the component. Turning it on and off far more than it is intended to do will shorten it's lifespan, keeping it running unnecessarily will shorten it too. Pushing it to it's limits will shorten the lifespan, and pushing it OVER the limits will shorten the lifespan considerably...

Maintaining the component as indicated by the manufacturer will in it's turn actually increases it's lifespan, except when the maintenance is performed in a rough or unprofessional manner...

Which one of these factors reduces or increases the lifespan the most, is usually unknown, and topic for a lot of heated discussions...

So, to make a long story even longer: MTBF is a number (usually representing hours) of time that gives you an idea of how long a component will work, before wear and tear may noticably start to degrade it's performance...

Really doesn't matter - if the MTBF is too high, then your system will be obsolete while it's still running. So you'll need a new one anyway!