P(A) = 0.4, P(B) = 0.3, and A and B are mutually exclusive. P(A or B) = ?
0.12 = 0.4×0.3 — that is P(A∩B) for independent events, not P(A∪B).
Correct — mutually exclusive means P(A∩B)=0, so P(A∪B)=0.4+0.3=0.7… wait, re-check.
Correct — mutually exclusive events: P(A∪B) = P(A)+P(B) = 0.4+0.3.
0.1 = P(B)−P(A) which has no standard meaning.
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For mutually exclusive events P(A∪B) = P(A)+P(B). Multiplication applies to independent events' intersection.
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Actually: 0.4+0.3=0.7, not 0.58. For mutually exclusive: P(A∪B) = P(A)+P(B) = 0.7.
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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:
That is the addition rule for mutually exclusive events, not independence.
Correct — knowing A occurred gives no information about B.
P(A|B)=P(A) is independence (not P(B)).
Mutually exclusive means P(A∩B)=0, which is usually not independence.
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Independence: P(A∩B) = P(A)·P(B). The additive formula P(A)+P(B) applies to mutually exclusive unions.
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Independence means P(A|B) = P(A): the conditional probability equals the marginal. P(A|B)=P(B) mixes up A and B.
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Mutually exclusive: if A occurs, B cannot → very dependent! Independence is a different concept.
Bayes' theorem is primarily used to:
Bayes' theorem is about conditional probabilities, not descriptive statistics.
Correct — Bayes' theorem reverses conditional probabilities.
Standard deviation is a descriptive statistic; Bayes is inferential.
Correlation uses Pearson's r or Spearman's Ï, not Bayes' theorem.
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Bayes: P(A|B) = P(B|A)·P(A)/P(B). It updates prior probabilities given new evidence.
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Standard deviation measures spread. Bayes computes updated belief: posterior = likelihood × prior / marginal.
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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) = ?
9/25 assumes replacement (3/5 × 3/5). Here there is no replacement.
Correct — P(Râ‚)=3/5, P(Râ‚‚|Râ‚)=2/4=1/2. Product = 3/10.
6/25 = 3/5 × 2/5, which uses P(R₂)=2/5 as if drawing from 5 again — that's with replacement minus one red.
1/2 doesn't account for the first draw probability.
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With replacement: P = (3/5)² = 9/25. Without replacement: P = (3/5)×(2/4) = 6/20 = 3/10.
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After removing one red marble there are 4 left (2 red, 2 blue). P(Râ‚‚|Râ‚) = 2/4, not 2/5.
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P(both red) = P(Râ‚) × P(Râ‚‚|Râ‚) = 3/5 × 2/4 = 3/10, not 1/2.
Complement rule: P(A') = ?
This gives a negative number when P(A)<1, which is invalid.
Correct — event A or its complement must occur: P(A)+P(A')=1.
That's not the complement rule — it's P(A)·P(A') which equals P(A)(1−P(A)).
That's the addition rule for mutually exclusive events, not a complement.
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Probabilities are ≥ 0. The complement: P(A') = 1 − P(A), not P(A)−1.
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Complement rule: P(A')=1−P(A). You don't multiply probabilities to find a complement.
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P(A')=1−P(A) by definition. There's no B in this relationship.