milky way - visualisation of galaxy density

For your experiment, you can take it 2D, so let's ingnore the thickness of Milky Way.

So, we have 400 billions stars(the real number is between 100B-400B) in a circle with 100k ly(the real number is between 100k-160k).

The area of the circle is pi*50k*50k ~= 7800M sqly = 7.8 billions sqly.

We have then ~51 randomly stars per square ly.

But you do not sting the Milky Way with your stick, but cut through it.

If you swipe through the middle, you have 51*100k stars in a width of 1 ly.

So, you have a chance to hit a star per 1/5.100.000 ly which means 1.855.000km.

But the diameter of sun(of your stick) is 1.395.000km. Taking diameter of sun as medium diameter of a star(this is not true), you have a chance of 1.4/1.8 = 77% to hit a star. If you go with the "low density" numbers, 100B stars and 160k ly diameter, you'll get ~31%.

My conclusion is you often hit a star.