Independent Events and Unions of Events - AP Statistics

1. What is the probability of the empty set?

   $P(\text{empty set}) = 0$. No outcomes exist in the empty set.

2. What is the intersection of events $A$ and $B$?

   The set of outcomes in both $A$ and $B$. Outcomes common to both events.

3. If $P(A) = 0.2$ and $P(B) = 0.4$, calculate $P(A \text{ and } B)$ for independent events.

   $P(A \text{ and } B) = 0.2 \times 0.4 = 0.08$. Multiply the individual probabilities for independence.

4. If $P(A) = 0.3$ and $P(B) = 0.5$, find $P(A \text{ and } B^c)$ if independent.

   $P(A \text{ and } B^c) = 0.3 \times 0.5 = 0.15$. Apply independence rule with complement.

5. If $A$ and $B$ are independent, what is $P(A^c \text{ and } B^c)$?

   $P(A^c \text{ and } B^c) = (1 - P(A))(1 - P(B))$. Both complements occur independently.

6. For events $A$ and $B$, state the addition rule for probability.

   $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. The general formula for union probability.

7. Are events $A$ and $B$ independent if $P(A) = 0.3$, $P(B) = 0.4$, $P(A \text{ and } B) = 0.12$?

   Yes, because $0.3 \times 0.4 = 0.12$. The product equals the intersection probability.

8. Find $P(A \text{ or } B^c)$ given $P(A) = 0.3$, $P(B) = 0.4$, independent events.

   $P(A \text{ or } B^c) = 1 - 0.3 \times 0.4 = 0.88$. Use complement rule: $1 - P(A^c \text{ and } B)$.

9. Find $P(A^c)$ given $P(A) = 0.75$.

   $P(A^c) = 1 - 0.75 = 0.25$. Apply the complement rule directly.

10. State the formula for $P(A \text{ or } B)$ for any events $A$ and $B$.

    $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General addition rule accounting for overlap between events.

11. What is the probability of $A$ if $P(A \text{ and } B) = 0.2$ and $P(B) = 0.5$, for independent events?

    $P(A) = \frac{0.2}{0.5} = 0.4$. Divide intersection by the other event's probability.

12. For independent events, how to find $P(A \text{ and } B^c)$?

    $P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Multiply probability of $A$ by complement of $B$.

13. Find $P(A \text{ or } B)$ if $P(A) = 0.4$, $P(B) = 0.5$, $P(A \text{ and } B) = 0.2$.

    $P(A \text{ or } B) = 0.4 + 0.5 - 0.2 = 0.7$. Standard addition rule application.

14. What does $P(A \text{ and } A^c)$ equal?

    $P(A \text{ and } A^c) = 0$. An event cannot occur with its complement.

15. What does it mean if $P(A \text{ and } B) \neq P(A) \times P(B)$?

    Events $A$ and $B$ are not independent. The product rule for independence fails.

16. Which rule applies to independent events for conditional probability?

    $P(B|A) = P(B)$. Knowing $A$ doesn't change probability of $B$.

17. If $P(A) = 0.6$, $P(B \text{ and } A) = 0.24$, find $P(B)$ if independent.

    $P(B) = \frac{0.24}{0.6} = 0.4$. Use independence: divide intersection by $P(A)$.

18. What is the probability of a sure event?

    $P(\text{sure event}) = 1$. Certain events have maximum probability.

19. If $P(A \text{ or } B) = 0.9$ and $P(A) = 0.6$, $P(B) = 0.5$, find $P(A \text{ and } B)$.

    $P(A \text{ and } B) = 0.6 + 0.5 - 0.9 = 0.2$. Rearrange the addition rule to solve for intersection.

20. Define a union of two events $A$ and $B$.

    The set of outcomes in $A$, $B$, or both. All outcomes where at least one event occurs.

21. Define mutually exclusive events.

    Events that cannot occur simultaneously. Their intersection has zero probability.

22. What is the probability of $A \text{ and } B$ for independent events?

    $P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.

23. What is the probability of $A^c$ and $A$ occurring together?

    $P(A^c \text{ and } A) = 0$. Complements are mutually exclusive.

24. What probability distribution is used for independent trials?

    Binomial distribution. Used for repeated independent trials with fixed success probability.

25. If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \text{ or } B) = 0.7$, are $A$ and $B$ independent?

    No, because $0.4 \times 0.5 \neq 0.2$. Check if $P(A \text{ and } B) = P(A) \times P(B) = 0.2$.

26. State the complement rule for event $A$.

    $P(A^c) = 1 - P(A)$. Probability of an event plus its complement equals 1.

27. What is $P(A^c \text{ or } B^c)$ if $A$ and $B$ are independent?

    $P(A^c \text{ or } B^c) = 1 - P(A)P(B)$. Use De Morgan's law: complement of intersection.

28. If $P(A) = 0.3$, $P(B) = 0.4$, find $P(A^c \text{ and } B^c)$ for independent events.

    $P(A^c \text{ and } B^c) = 0.7 \times 0.6 = 0.42$. Calculate complements first, then multiply.

29. What is the probability of the union of $A$ and $B$ if they are mutually exclusive?

    $P(A \text{ or } B) = P(A) + P(B)$. No overlap means no subtraction needed.

30. What is $P(A^c \text{ or } B)$ if $A$ and $B$ are independent?

    $P(A^c \text{ or } B) = 1 - P(A) \times (1 - P(B))$. Use De Morgan's law and independence property.

31. Find $P(A \text{ or } B)$ given $P(A) = 0.5$, $P(B) = 0.3$, $P(A \text{ and } B) = 0.1$.

    $P(A \text{ or } B) = 0.5 + 0.3 - 0.1 = 0.7$. Apply the addition rule: sum minus overlap.

32. If events $A$ and $B$ are independent, what is $P(A \text{ and } B^c)$?

    $P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Event $A$ occurs while $B$ doesn't.

33. What is $P(A \text{ or } A^c)$?

    $P(A \text{ or } A^c) = 1$. An event and its complement cover all possibilities.

34. What is the definition of independent events?

    Events where $P(A \text{ and } B) = P(A) \times P(B)$. The occurrence of one doesn't affect the other's probability.

35. How do you verify if two events are independent?

    Check if $P(A \text{ and } B) = P(A) \times P(B)$. Apply the independence condition test.

36. If $A$ and $B$ are mutually exclusive, what is $P(A \text{ and } B)$?

    $P(A \text{ and } B) = 0$. Mutually exclusive events cannot both occur.

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