Question

What is the chance of going against Johanna in a contest?

Answer 1

Unlike previous mothers, Johanna is less content to stay at home.
After the player begins their journey, she can be found at the Contest
Hall and occasionally enters Master Rank Super Contests with her
Kangaskhan, Jumpy

http://bulbapedia.bulbagarden.net/wiki/Johanna

So there is no exact percent, but occasionally would be the answer.

Answer 2

She occasionally enters her pokemon Jumpy (Kangaskhan) in Master Rank Super Contests