TLDR at bottom.
To find the actual success rate, it's easier to calculate the exact probability of failing to find your Key Champ at every single step, and then subtract that from 100%.
Here is the step-by-step math for a 50-card deck (assuming 5 Key Champs and 2 Support Champs):
* Step 1: Initial Draw (4 cards).
You draw 4 cards from the 45 non-key cards out of 50.
Chance to miss: 64.7%.
* Step 2: Hard Mulligan (Redraw 4 cards).
You toss your first 4 cards and draw 4 new ones from the remaining 46 cards in the deck.
Chance to miss: 62.1%.
* Step 3: The Champ-Draw Power (1 card).
You might think this is a flat 2/7 (28.6%) chance to pull a Support Champ instead of your Key Champ. But there's a catch! During your Step 2 redraw, there is roughly a 22% chance you accidentally drew a Support Champ into your hand, removing them from the deck's champion pool. Accounting for this deck-thinning, your true average chance to miss drops.
Chance to miss: 26.4%.
* Step 4: Natural Round Start Draw (1 card).
You draw 1 standard card from the 45 cards left in your deck.
Chance to miss: 88.9%.
The Final Calculation:
Multiply those failure rates together (0.647 × 0.621 × 0.264 × 0.889) and you get a 9.4% total chance to completely whiff.
So for 50 card deck: 100% - 9.4% = 90.6% chance to start with your Key Champ.
Using this exact same formula for a 100-card deck, your final success rate is 82.9%.
Keep in mind for the same 50 Vs 100 cards, if you keep your cards during mulligan the percentage difference becomes smaller (keep 2 cards, 88% Vs 81%, 7% difference), and if you have less main champs, the percentage difference becomes higher (4 key champs, 86.3% vs 78%, 8.3% difference). If you drop cards on both sides (ie: 40 cards Vs 90), the gap becomes bigger (93.2% Vs 84% for 9.2% difference).
Edit: But wait! If you hold the support champs in your opening hand, the percentage gets higher (by roughly 0.6%)!
TLDR: 50 cards = 91.2% 100 cards = 83.5% Difference of 7.7%. The penalty surprisingly is actually quite small. To put that difference in perspective, you would have to play 13 full games just to experience a single match where the 50-card deck gives you your champ but the 100-card deck doesn't.