On a calculator, there are numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any key. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last push. The Scattered Scientist pressed a lot of buttons in a random sequence. Find approximately the probability with which the outcome of the resulting chain of actions is an odd number?