If you really need a card on your starting hand to win, and otherwise you lose, then you have to take mulligans until the card shows up or you run out of mulligans. The chance to have Treasure Hunt in your hand by turn 2 with at 6 mulligans is 87.5%
When the alternative to taking a mulligan is losing the game, then taking another mulligan that draws you at least 1 card is always the right choice. Basically, you stop taking mulligans until you find the card you need, or you are down to 1 hand card.
Therefore, calculating mulligan probabilities is somewhat besides the point. If you still want to know the odds, then in the top answer to the question How do you calculate the likelihood of drawing certain cards in your opening hand? there is a link to a calculator that you can use to calculate a hypergeometric distribution.
You put in the following numbers, from top to bottom:
Population size: 60 (for the deck size),
Number of successes in population: 4 (number of copies of the card you need),
Sample size: 7 (hand size, adjust for mulligans), and
Number of successes in sample (x): 1 (you want at least 1 copy in hand).
Then you can calculate the odds for any mulligan. On the play on your initial hand, you have a ~60% chance to NOT have the card in hand. After 1 mulligan, the chance is ~64% and so on. To calculate the odds of NOT having the card after n mulligans, you calculate those individual n mulligan probabilities, and multiply them. For example, after 2 mulligans, the chance to not have the card is 0.6*0.64 = 38.4%, so the chance to have the card is 100%-38.4% = 61.6%.
Since you can cast Treasure Hunt only on turn 2, you can effectively consider the last mulligan to be 1 card larger than actual, because you will draw another card before you really need the Treasure Hunt in your hand. So for example, if you take 1 mulligan and keep the next hand regardless, then the probability of having the card in your hand by turn 2 is really 1 - .6*.6 = 64%.
The maximum chance to see the card is achieved by taking all mulligans down to 1 card. The probabilities to NOT have the card after each starting hand draw are, in order:
60%, 64.8%, 69.9%, 75.3%, 81%, 87%, 93.3%
So the chance to NOT have the card in hand after 6 mulligans is:
0.6 * 0.648 * 0.699 * 0.753 * 0.81 * 0.87 * 0.933 = 13.5%
The chance to draw at least 1 copy after 6 mulligans is
1 - 13.5% = 86.5%
As discussed before, the last mulligan is effectively 1 card larger:
0.6 * 0.648 * 0.699 * 0.753 * 0.81 * 0.87 * 0.87 = 12.5%
For a total chance to draw the card by turn 2 after 6 mulligans:
1 - 12.5% = 87.5%
Note that since you can scry 1 after your last mulligan means that you can practically eliminate another unwanted card from your deck, which makes the probability a little bit higher still.