Hearthstone is a free to play online card game. The basics of it involve one of 9 classes with a 30 card deck with a max of two unique cards except for a legend cards which have a max of one. At the start of each game, a player draws 3 or 4 cards depending who plays first. A player can either keep the cards dealt or mulligan any of the dealt cards and draw the same amount back. If a player chooses to mulligan a card, the card is removed from the deck and reshuffled back in, after the mulligan phase is complete. After this point, cards are drawn one at a time at the start of each turn. As well as drawing a card, a player gains one more mana crystal that can be spent to play cards that turn, up to a maximum of 10 crystals. The game continues until a player reaches 0 life from the started 30.
In Hearthstone there are certain decks that benefit greatly from having at least 1 of the cards in the opening hand. In the past this was Undertaker played in a deathrattle hunter deck. Before the nerf, this card could gain attack and health very quickly as the player played more deathrattle minions to buff it up. In recent times, the Mechwarper which allows other cards of the mech type to cost 1 less have become popular. This effectively ramps the mech player to more expensive cards quicker. On the flip side, a warrior class player may want to mulligan hard for a Fiery War Axe to be able to remove early high priority minions, such as the Mechwarper.
For this, it is assumed that the player has two copies of the card in the deck. To calculate the probability of drawing a number of cards and getting a specific card can be done using combinations. To do this, calculating the probability of NOT drawing the card is calculated such that: P(drawing card) = 1 - P(NOT drawing the card).
First Player - 3 Card Mulligan
Card in First 3
The probability of getting the card is in the first 3 becomes:
P(Card in First 3) = 1 - P(Card NOT in opening) = 1 - # of combinations / # of total combinations
##  0.1931
Card After Mulligan
From this, calculating the the probability of getting the card in the second set, assuming not keeping any of the original 3:
P(Card after Mulligan) = 1 - (P(Card NOT in First 3) * P(Card NOT After First 3))
##  0.3655
Since the original three cards can not be drawn in the mulligan, it reduces the total number of cards to 27.
Card After Mulligan AND first draw
With only one card being drawn, this becomes:
P(Card after Mulligan AND First Draw) = 1 - (P(Card NOT in First 3) * P(Card NOT in Mulligan)) * P(NOT drawing first)
##  0.4125
The 27 comes from the mulliganed cards being re-shuffled back into the deck after the mulliganing phase is over.
Second Player - 4 Card Mulligan
Card after Mulligan AND first draw
Following the logic of the first player:
##  0.5098
From this, if a player hard mulligans for a card the probability of obtaining it are:
First Player: 41.3%
Second Player: 51.0%