IPhone 4 :: Turns Off Randomly While Using Calculator Or Playing Music

I have an iPhone 4, and it is probably 3-4 years old, and has no obvious extrernal problems except that the lock screen button doesn't work. That stopped working about 3 months ago, and i went to the apple store and they said there was nothing to do about it. They also looked at the iPhone and said that there were no problems with the battery life or anything (it was a dot in the greenest section on the picture for battery life). but, since then my iPhone had been randomly turning off. At the beginning, since my lock button was broken i would just plug it in, and it would turn on about 2 mins later saying very low battery, but it would go straight back to whatever charge it was before in a couple of minutes (usually about 50%). Â

But today, it randomly turned off while i was using the calculator. I tried pressing the home button, but nothing happened. I plugged it in, it made the 'plm' sound it does when it charges, but the screen turned black. Â

I think it was maybe a problem with the display. Because i have the apple headphones, and i used them to make the music play, and it did, i just couldnt see the screen. About 3 hours after it turned off (no screen on, but still music and sound), it turned back on again fully (with the screen). I have already been to the apple store about this matter, and they said they essentially have no idea why its happening. Usually it turns back on when it plugs in, but today it didnt.Â

Info:
iPhone 4