Can you get any information from your own mulligan?
Solution 1:
After the mulligan, the deck is shuffled again for the match. So, no there is no way to guess when you actually draw a card.
Taken from the Hearthstone Wiki with a link to Ben Brode (Twitter), Senior Game Designer on Hearthstone at Blizzard Entertainment.
They are however returned to the deck after the mulligan is complete, to be drawn during the following match. The remaining deck is shuffled or randomised once the mulligan is complete