there is not enough data to tell you what you need.
bipolar transistors are 'turned on' by current, that means that the more current you use to drive it, the more current it will let to pass.
if you want to turn on a transistor with 3mA, it can be possible, but it depends on the gain of it. i am not sure about the gain of the 2222 but i tiink it is around 100 or greater. anyway, the current it will let to pass is equalt to the product between the gain and the current you use to open it. if i say that the gain is 100 and the base emitter current is 3 mA then, the current that will pass trough the load is 300mA or less, depending on the load.