Study Probability Fundamentals — complete the practice problems in your notebook.
P(A) = 0.4, P(B) = 0.3, and A and B are mutually exclusive. P(A or B) = ?
For mutually exclusive events P(A∪B) = P(A)+P(B). Multiplication applies to independent events' intersection.
Actually: 0.4+0.3=0.7, not 0.58. For mutually exclusive: P(A∪B) = P(A)+P(B) = 0.7.
Subtraction of probabilities isn't a standard operation. Use the addition rule: P(A∪B)=P(A)+P(B)−P(A∩B).
Two events are independent if:
Independence: P(A∩B) = P(A)·P(B). The additive formula P(A)+P(B) applies to mutually exclusive unions.
Independence means P(A|B) = P(A): the conditional probability equals the marginal. P(A|B)=P(B) mixes up A and B.
Mutually exclusive: if A occurs, B cannot → very dependent! Independence is a different concept.
Bayes' theorem is primarily used to:
Bayes: P(A|B) = P(B|A)·P(A)/P(B). It updates prior probabilities given new evidence.
Standard deviation measures spread. Bayes computes updated belief: posterior = likelihood × prior / marginal.
Correlation measures linear association. Bayes updates conditional probabilities.
A bag has 3 red and 2 blue marbles. You draw one (not replaced), then draw again. P(both red) = ?
With replacement: P = (3/5)² = 9/25. Without replacement: P = (3/5)×(2/4) = 6/20 = 3/10.
After removing one red marble there are 4 left (2 red, 2 blue). P(R₂|R₁) = 2/4, not 2/5.
P(both red) = P(R₁) × P(R₂|R₁) = 3/5 × 2/4 = 3/10, not 1/2.
Complement rule: P(A') = ?
Probabilities are ≥ 0. The complement: P(A') = 1 − P(A), not P(A)−1.
Complement rule: P(A')=1−P(A). You don't multiply probabilities to find a complement.
P(A')=1−P(A) by definition. There's no B in this relationship.