If you haven't yet read part one of When to Mulligan, you can read it here.
How your Deck affects how you Mulligan
This matters most if you are a combo deck or a very fast deck. If you are a combo deck, you need to know if you will be able to combo off by your target turn. Your probability to draw any one card from a 60 card deck assuming you have four copies of it are as follows:
First Turn: ~6.67%
Second Turn: ~13.33%
Third Turn: ~20%
Fourth Turn: ~26.67%
If you're missing one combo piece you'll only draw it by your fourth turn one quarter of the times you take that risk. That is why combo decks are hard to make work without significant amounts of card draw or tutors.
How your Opponent's Deck affects how you Mulligan
This matters most in games two and three. After you know what your opponent's deck is, you have to take it into consideration when you mulligan. If your opponent is playing an aggressive deck, can you stop them? If you can't, you probably should mulligan. Is your opponent playing combo, if so, can you stop them? If not, that can be another sign that you should mulligan.
If your opponent plays a deck with a lot of either targeted discard or targeted removal, can you survive a Thoughtseize taking one of your cards or a Path killing one of your creatures? If not, then a mulligan might also be in order.
A good way to sum it up is: If you can't stop what your opponent's doing, or your opponent can easily stop what you're doing, you should mulligan.
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