No, you are wrong. Of course this is not simple, but lets take an exmple:
Lets say after 1st bullet there's 1/3 chance of hitting middle, off left, off right. System guarantees that during spraying, the same position is not randomized right after each other to ensure that odds don't greate 5 middle hits in a row (would be unfair).
If you keep mouse in the middle stationary, 2nd bullet has chances of 1/3. If your 2nd one hits, the 3rd one won't hit!
If you move your mouse after 1st bullet constantly, your 2nd bullet will have same 1/3 of chances to hit. If you hit with 2nd bullet and move your mouse for 3rd one, it will have 1/2 chances to hit! If you do not hit with that one, and move your mouse, you still have chance to hit. Where as staying stationary you ensure that the following bullet won't hit.
Check the legendary "should I switch" experiment from Mythbusters. There you have three doors and behind one is the price. You select one, and they reveal one of the wrong doors and ask you, do you want to change your choice. Now, you always should change your choice in this situation! You will get 1/2 chances to have correct door, instead of the original 1/3.