Probability Rules — Practice Quiz

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

Practice quiz

1. Given $P(X) = 0.6$, $P(Y) = 0.5$, and $P(X \cap Y) = 0.3$, what is $P(X \cup Y)$?

   • $0.8$
   • $0.9$
   • $0.7$
   • $0.6$

   Answer: $0.8$

2. Events $M$ and $N$ are mutually exclusive. If $P(M) = 0.45$ and $P(N) = 0.3$, what is $P(M \cup N)$?

   • $0.15$
   • $0.75$
   • $0.6$
   • $0.9$

   Answer: $0.75$

3. If $P(A) = 0.7$ and $P(B|A) = 0.6$, what is $P(A \cap B)$?

   • $0.42$
   • $0.1$
   • $0.7$
   • $0.6$

   Answer: $0.42$

4. Events $E$ and $F$ are independent. If $P(E) = 0.8$ and $P(F) = 0.25$, what is $P(E \cap F)$?

   • $0.2$
   • $1.05$
   • $0.55$
   • $0.8$

   Answer: $0.2$

5. For two events $A$ and $B$ to be mutually exclusive, which of the following must be true?

   • $P(A \cap B) = P(A)P(B)$
   • $P(A \cap B) = 0$
   • $P(A \cup B) = 1$
   • $P(A|B) = P(A)$

   Answer: $P(A \cap B) = 0$

6. If events $C$ and $D$ are independent, which of the following statements is always true?

   • $P(C \cup D) = P(C) + P(D)$
   • $P(C|D) = 0$
   • $P(C \cap D) = P(C)P(D)$
   • $P(C) + P(D) = 1$

   Answer: $P(C \cap D) = P(C)P(D)$

7. In a class, $60\%$ of students like Math ($M$) and $40\%$ like Science ($S$). If $20\%$ like both, what percentage of students like Math or Science?

   • $100\%$
   • $80\%$
   • $60\%$
   • $40\%$

   Answer: $80\%$

8. A bag contains $5$ red balls and $3$ blue balls. If two balls are drawn without replacement, what is the probability that both are red?

   • $\frac{25}{64}$
   • $\frac{15}{56}$
   • $\frac{20}{64}$
   • $\frac{20}{56}$

   Answer: $\frac{20}{56}$

9. Which statement is true about mutually exclusive and independent events?

   • Mutually exclusive events are always independent.
   • Independent events are always mutually exclusive.
   • Events cannot be both mutually exclusive and independent unless one event has probability $0$.
   • If $P(A) > 0$ and $P(B) > 0$, then mutually exclusive events are never independent.

   Answer: If $P(A) > 0$ and $P(B) > 0$, then mutually exclusive events are never independent.

10. Given $P(A) = 0.5$, $P(B) = 0.4$, and $P(A \cup B) = 0.7$. Are events $A$ and $B$ independent?

    • Yes, because $P(A \cap B) = P(A)P(B)$.
    • No, because $P(A \cap B) \neq P(A)P(B)$.
    • Yes, because $P(A \cup B) = P(A) + P(B)$.
    • No, because $P(A \cup B) \neq P(A) + P(B)$.

    Answer: Yes, because $P(A \cap B) = P(A)P(B)$.