Rules: not, or, and

Three rules let you combine probabilities instead of counting every outcome.

Not: the complement

P(not A) = 1 − P(A). It is often easier to find the chance that something never happens. "At least one six in four rolls" is 1 − P(no six in four rolls).

And: independent events multiply

Two events are independent when one tells you nothing about the other, like two separate dice. Then P(A and B) = P(A) × P(B). So P(no six in four rolls) = (5/6)⁴.

Or: add, then remove the overlap

P(A or B) = P(A) + P(B) − P(A and B). Adding alone counts outcomes in both events twice. Drawing one card: P(king or heart) = 4/52 + 13/52 − 1/52, because the king of hearts is in both.

from fractions import Fraction

no_six = Fraction(5, 6) ** 4
print("P(no six in 4 rolls) =", no_six)
print("P(at least one six) =", 1 - no_six, "≈", round(float(1 - no_six), 3))

king, heart, both = Fraction(4, 52), Fraction(13, 52), Fraction(1, 52)
print("P(king or heart) =", king + heart - both)

Common mistakes

• Adding probabilities of events that overlap (you get more than the truth, sometimes more than 1).
• Multiplying probabilities of events that are not independent, such as two cards drawn without putting the first back. That needs conditional probability, the next lesson.

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