Are these the only candidates for the position of manager?
Assuming they are then..
Turn the ratio into a fraction/decimal/percentage/whatever you like working with probability in. So 4+2+3=9. So X has 4/9ths of a chance to be manager, Y 2/9, Z 3/9.
So the probability of X becoming manager is 4/9 or 44.44 %
Then your method for finding the probability of the bonus seems to be correct, but you have to change the P(X), P(Y), and P(Z) in a similar fashion to what we've just done here..