Probability Rules — Hard Practice Quiz

A Probability cheat sheet for Probability Rules — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Addition Rule (General)

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Addition Rule (Mutually Exclusive)

$$P(A \cup B) = P(A) + P(B)$$

Where: When \(P(A \cap B) = 0\)

Multiplication Rule (General)

$$P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$$

Multiplication Rule (Independent)

$$P(A \cap B) = P(A) \cdot P(B)$$

Practice quiz

  1. Given $P(A) = 0.4$, $P(B) = 0.7$, and $P(B|A) = 0.5$. Find $P(A \cup B)$.

    • $0.9$
    • $1.1$
    • $0.82$
    • $0.2$

    Answer: $0.9$

  2. Events $A$ and $B$ are independent. If $P(A) = 0.6$ and $P(B) = 0.5$, what is $P(A \cup B)$?

    • $0.8$
    • $1.1$
    • $0.3$
    • $0.7$

    Answer: $0.8$

  3. Given $P(A) = 0.3$, $P(B) = 0.6$, and $P(A \cup B) = 0.7$. Find $P(A|B)$.

    • $\frac{1}{3}$
    • $\frac{2}{3}$
    • $0.2$
    • $0.18$

    Answer: $\frac{1}{3}$

  4. Can two events $A$ and $B$ with $P(A) > 0$ and $P(B) > 0$ be both independent and mutually exclusive?

    • No, never.
    • Yes, always.
    • Yes, if $P(A) + P(B) = 1$.
    • Yes, if $P(A) = P(B)$.

    Answer: No, never.

  5. Given $P(A) = 0.4$, $P(A \cup B) = 0.7$, and $P(B|A) = 0.5$. Find $P(B)$.

    • $0.5$
    • $0.2$
    • $0.4$
    • $0.7$

    Answer: $0.5$

  6. If events $A$ and $B$ are mutually exclusive and $P(B) > 0$, what is $P(A|B)$?

    • $0$
    • $P(A)$
    • $1 - P(B)$
    • $P(A) \cdot P(B)$

    Answer: $0$

  7. In a certain population, $P(\text{smoker}) = 0.25$, $P(\text{has lung disease}) = 0.15$. The probability that a person is a smoker AND has lung disease is $0.1$. Are smoking and lung disease independent events? What is the probability that a person is a smoker OR has lung disease?

    • No, $P(\text{smoker or lung disease}) = 0.3$.
    • Yes, $P(\text{smoker or lung disease}) = 0.3$.
    • No, $P(\text{smoker or lung disease}) = 0.4$.
    • Yes, $P(\text{smoker or lung disease}) = 0.375$.

    Answer: No, $P(\text{smoker or lung disease}) = 0.3$.

  8. Given $P(A) = 0.4$, $P(B) = 0.3$, and $P(A|B) = 0.6$. Find $P(B|A)$.

    • $0.45$
    • $0.6$
    • $0.18$
    • $0.3$

    Answer: $0.45$

  9. If $A$ and $B$ are independent events, and $P(A) = 0.2$, $P(B) = 0.3$. What is the probability that neither $A$ nor $B$ occurs?

    • $0.56$
    • $0.44$
    • $0.5$
    • $0.7$

    Answer: $0.56$

  10. Events $X$ and $Y$ have probabilities $P(X) = 0.6$, $P(Y) = 0.5$. If $P(X \cup Y) = 0.8$, determine if $X$ and $Y$ are independent and if they are mutually exclusive.

    • Independent, not mutually exclusive.
    • Not independent, not mutually exclusive.
    • Independent, mutually exclusive.
    • Not independent, mutually exclusive.

    Answer: Independent, not mutually exclusive.

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