1-draw is probably a simpler problem to tackle... it is also expected to have more winning games
The "thoughtful" variant

https://archive.md/DxAB
February 9, 1999
Persi Diaconis (Stanford University)
Mathematics of Solitaire
Abstract:
One of the embarrassments of applied probability is that we can not analyze the original game of solitaire. I will show that a simplified solitaire leads to fascinating mathematics and complete analysis. The mathematics involves combinatorics, random matrices, and Riemann-Hilbert theorem. This is a joint work with David Aldous.

This colloquium talk is accessible to a general audience including undergraduate students.