How to prove the average case time complexity of bubble sort and cocktail shaker sort? (use the number of comparisons as basic operation)

I understand how to prove the best case and worst case of these algorithms e.g. bubble sort best case is exactly n-1 operations and worst case is n(n-1)/2 comparisons. But how do I show the average case?

Note: I am using the version of bubble sort that exits if no swaps are performed for the current pass of the algorithm