Conditional Probability - AP Statistics

1. Determine $P(A|B)$ if $P(A \cap B) = 0.28$ and $P(B) = 0.7$.

   $P(A|B) = 0.4$. Using $P(A|B) = \frac{0.28}{0.7} = 0.4$.

2. Identify the type of probability: $P(A|B)$.

   Conditional Probability. Probability of $A$ given that $B$ has occurred.

3. What does $P(A|B) = 0$ imply?

   Event $A$ cannot occur if $B$ has occurred. Event $A$ is impossible when $B$ has already happened.

4. Identify $P(B|A)$ if $P(A \cap B) = 0.2$ and $P(A) = 0.4$.

   $P(B|A) = 0.5$. Using $P(B|A) = \frac{0.2}{0.4} = 0.5$.

5. What is the inverse relationship of $P(A|B)$?

   $P(B|A)$ using Bayes' Theorem. The probability of $B$ given $A$ has occurred.

6. Determine $P(B|A)$ if $P(A|B)=0.5$, $P(B)=0.4$, $P(A)=0.2$.

   $P(B|A) = 1$. Using Bayes' Theorem: $\frac{0.5 \times 0.4}{0.2} = 1$.

7. If $P(A|B)=0.2$, $P(B)=0.6$, find $P(A \cap B)$.

   $P(A \cap B) = 0.12$. Using multiplication rule: $0.2 \times 0.6 = 0.12$.

8. If $P(A|B)=0.6$ and $P(B)=0.5$, find $P(A \cap B)$.

   $P(A \cap B) = 0.3$. Using multiplication rule: $0.6 \times 0.5 = 0.3$.

9. Which rule relates joint probability to conditional probability?

   Multiplication Rule. Connects joint and conditional probabilities through multiplication.

10. Express the multiplication rule for probability in terms of $P(A|B)$.

    $P(A \cap B) = P(A|B) \cdot P(B)$. Rearranges conditional probability formula to find joint probability.

11. Calculate $P(A|B)$ if $P(A \cap B) = 0.4$ and $P(B) = 0.8$.

    $P(A|B) = 0.5$. Using $P(A|B) = \frac{0.4}{0.8} = 0.5$.

12. What does $P(A|B) = 1$ imply about the events?

    Event $A$ occurs whenever $B$ occurs. Event $A$ is certain to happen when $B$ has occurred.

13. What does $P(A \cap B) = 0$ indicate?

    Events $A$ and $B$ are mutually exclusive. Events cannot occur simultaneously; they are disjoint.

14. What is the complementary rule for conditional probability?

    $P(A^c|B) = 1 - P(A|B)$. Complement of conditional probability within the same condition.

15. Identify $P(A|B)$ if $P(A \cap B) = 0.3$ and $P(B) = 0.6$.

    $P(A|B) = 0.5$. Using $P(A|B) = \frac{0.3}{0.6} = 0.5$.

16. What is the definition of conditional probability?

    Probability of an event given another has occurred. The likelihood of event $A$ happening when we know $B$ has occurred.

17. What is the probability of $A$ given $A$ and $B$ are disjoint?

    $P(A|B) = 0$. Disjoint events have zero conditional probability.

18. Determine if $P(A|B) = 0.3$ and $P(A) = 0.5$ indicate independence.

    Not independent. Since $P(A|B) \neq P(A)$, the events are dependent.

19. What is $P(A \cap B^c)$ in terms of $P(A|B^c)$?

    $P(A \cap B^c) = P(A|B^c) \cdot P(B^c)$. Joint probability for event $A$ and complement of $B$.

20. Calculate $P(A|B)$ for $P(A \cap B) = 0.15$ and $P(B) = 0.3$.

    $P(A|B) = 0.5$. Using $P(A|B) = \frac{0.15}{0.3} = 0.5$.

21. What is $P(B \cap A)$ if $P(A \cap B) = 0.3$?

    $P(B \cap A) = 0.3$. Intersection is commutative; order doesn't matter.

22. If $P(A|B) = 0.4$, what is $P(A^c|B)$?

    $P(A^c|B) = 0.6$. Since $P(A|B) + P(A^c|B) = 1$.

23. Find $P(A|B)$ with $P(A \cap B) = 0.25$ and $P(B) = 0.5$.

    $P(A|B) = 0.5$. Using $P(A|B) = \frac{0.25}{0.5} = 0.5$.

24. Define $P(A \cap B)$ in terms of conditional probability.

    $P(A \cap B) = P(A|B) \cdot P(B)$. Joint probability expressed using conditional probability formula.

25. Define mutually exclusive events in terms of $P(A|B)$.

    $P(A|B) = 0$. Events that cannot occur together have zero conditional probability.

26. Find $P(A \cap B)$ for independent events $A$ and $B$. $P(A)=0.4$, $P(B)=0.5$.

    $P(A \cap B) = 0.2$. For independent events: $0.4 \times 0.5 = 0.2$.

27. What does $P(A|B) = P(A)$ signify about events $A$ and $B$?

    Events $A$ and $B$ are independent. The occurrence of $B$ doesn't affect the probability of $A$.

28. What does $P(B|A) = 1$ suggest?

    Event $B$ occurs whenever $A$ occurs. Event $B$ is certain to occur when $A$ has happened.

29. Determine if $P(A|B) = 0.2$ and $P(A) = 0.3$ indicate independence.

    Not independent. Since $P(A|B) \neq P(A)$, events are dependent.

30. State the formula for conditional probability.

    $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Divides joint probability by the probability of the conditioning event.

31. What is the condition for $P(A|B)=P(A|B^c)$?

    Events $A$ and $B$ are independent. Knowledge of $B$ or $B^c$ doesn't affect probability of $A$.

32. State the formula for $P(A^c \cap B)$ using conditional probability.

    $P(A^c \cap B) = P(B) - P(A \cap B)$. Probability of complement of $A$ intersecting with $B$.

33. State Bayes' Theorem.

    $P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$. Allows calculation of reverse conditional probability.

34. Evaluate independence: $P(A|B)=0.7$, $P(A)=0.7$.

    Independent. Since $P(A|B) = P(A)$, the events are independent.

35. What is the condition for two events to be independent?

    $P(A|B) = P(A)$ and $P(B|A) = P(B)$. Conditional probability equals marginal probability for independent events.

36. For independent events, what is $P(A \cap B)$?

    $P(A \cap B) = P(A) \cdot P(B)$. For independent events, joint probability is the product of marginals.

37. Identify $P(A|B)$ given $P(A \cap B) = 0.1$ and $P(B) = 0.25$.

    $P(A|B) = 0.4$. Using $P(A|B) = \frac{0.1}{0.25} = 0.4$.

38. Which theorem is used to find $P(B|A)$ from $P(A|B)$?

    Bayes' Theorem. Used to reverse conditional probability relationships.

Study these flashcards