Calculate draw odds without overcomplicating the game
Count helpful cards and cards left in your deck to estimate a draw. Use those odds to compare your plans.
One draw
Divide the number of helpful cards by the number of cards left.
If 12 cards remain and three can save you, your next normal draw has a 3 out of 12 chance. That is 25%, or one in four.
Count only cards that really solve the problem.
Two or more draws
After each draw, the deck gets smaller. The odds change.
Suppose two cards out of ten are outs. You can draw twice. The chance to miss the first time is 8/10. After a miss, it is 7/9 for the second draw.
Multiply those to find the chance to miss both: (8/10) × (7/9). Take that away from 1 to find the chance to get at least one out: about 37.8%.
A formula for larger counts
Let N be cards left, K be outs, and n be draws. The chance of at least one out is:
1 − C(N − K, n) / C(N, n)
C(a, b) counts ways to choose b items from a items. The math name is the hypergeometric model. You do not need to memorize it to use the simple examples above.
This assumes normal draws with no cards added or other changes. Tutors, special draws, and shuffled cards may need different math.
Starting hands work differently
In a mulligan, rejected cards are set aside while you get replacements. You can still get another copy. What you keep also changes the odds. Do not treat these as separate, normal deck draws.
Odds are part of the choice
A 40% chance to find an answer helps only if the answer gives you a good chance to win. Compare it with your other plan. During a turn, a useful rough estimate is often enough.