What's the probability to start a game of Vintage Dredge with Bazaar of Baghdad in your opening hand?
I approached this question numerically and drew one billion (1,000,000,000) sample hands.
My assumptions: The deck contains 60 cards, 4 of which are Bazaar of Baghdad and 4 are Serum Powder. No further assumptions regarding the deck building are required.

Then I draw the opening seven cards. If it contains at least 1 Bazaar, the hand is considered a hit. If I draw a Serum Powder, the hand is removed from the game and I draw the exact same number again. If I miss on both, Bazaar and Powder, than I take a mulligan (e.q. reshuffling the deck and drawing one card less).

This continues until I reach the point where I reshuffle a one-card-hand, since it was considered a miss and start the game with a zero-card-hand. In this situation I consider the somewhat new mulligan scry rule, which let's you scry one after mulligan. So even if I start with zero cards in hand, there is a chance that a Bazaar is in the top 2 cards of my library and (thanks to the scry) can be found within one draw step. That probability is also included in my end results.

My findings:


Probability of starting with a 7 card hand: 54.2 %
Probability of starting with a 6 card hand: 22.12 %
Probability of starting with a 5 card hand: 9.91 %
Probability of starting with a 4 card hand: 4.79 %
Probability of starting with a 3 card hand: 2.43 %
Probability of starting with a 2 card hand: 1.23 %
Probability of starting with a 1 card hand: 0.52 %
Probability of starting with a 0 card hand: 4.8 % (this means no Bazaar of Baghdad was found)
Probability of starting with a 0 card hand and Bazar on top (including scry 1): 0.9 %
Probability of complete failure: 3.9 %
Probability to hit: 95.2015011 %
Probability to hit with scry: 96.1029072 %


For the curious people, here is a plot how those probabilities converged: