AP Statistics Flashcards: Conditional Probability

What this deck covers

This deck focuses on Conditional Probability, giving you a quick way to review the definitions, rules, and examples that matter most for AP Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

Flashcard 3: Define $P(A \cap B)$ in terms of conditional probability.

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

Flashcard 4: What does $P(A|B) = 0$ imply?

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

Flashcard 5: Identify $P(B|A)$ if $P(A \cap B) = 0.2$ and $P(A) = 0.4$.

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

Flashcard 6: If $P(A|B) = 0.4$, what is $P(A^c|B)$?

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

Flashcard 7: What is the inverse relationship of $P(A|B)$?

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

Flashcard 8: Determine $P(B|A)$ if $P(A|B)=0.5$, $P(B)=0.4$, $P(A)=0.2$.

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

Flashcard 9: If $P(A|B)=0.2$, $P(B)=0.6$, find $P(A \cap B)$.

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

Flashcard 10: If $P(A|B)=0.6$ and $P(B)=0.5$, find $P(A \cap B)$.

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

Flashcard 11: Which rule relates joint probability to conditional probability?

Answer: Multiplication Rule. Connects joint and conditional probabilities through multiplication.

Flashcard 12: What is the inverse relationship of $P(A|B)$?

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

Flashcard 13: Identify $P(A|B)$ given $P(A \cap B) = 0.1$ and $P(B) = 0.25$.

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

Flashcard 14: Express the multiplication rule for probability in terms of $P(A|B)$.

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

Flashcard 15: Calculate $P(A|B)$ if $P(A \cap B) = 0.4$ and $P(B) = 0.8$.

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

Flashcard 16: What does $P(A|B) = 1$ imply about the events?

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

Flashcard 17: Determine $P(A|B)$ if $P(A \cap B) = 0.28$ and $P(B) = 0.7$.

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

Flashcard 18: What does $P(A \cap B) = 0$ indicate?

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

Flashcard 19: For independent events, what is $P(A \cap B)$?

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

Flashcard 20: What is the complementary rule for conditional probability?

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

Flashcard 21: Identify $P(A|B)$ if $P(A \cap B) = 0.3$ and $P(B) = 0.6$.

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

Flashcard 22: Define mutually exclusive events in terms of $P(A|B)$.

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

Flashcard 23: Express the multiplication rule for probability in terms of $P(A|B)$.

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

Flashcard 24: What is the definition of conditional probability?

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

Flashcard 25: What is the probability of $A$ given $A$ and $B$ are disjoint?

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

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

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

Flashcard 27: Determine if $P(A|B) = 0.3$ and $P(A) = 0.5$ indicate independence.

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

Flashcard 28: What is the condition for two events to be independent?

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

Flashcard 29: Identify $P(A|B)$ if $P(A \cap B) = 0.3$ and $P(B) = 0.6$.

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

Flashcard 30: State the formula for $P(A^c \cap B)$ using conditional probability.

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

Flashcard 31: What is $P(A \cap B^c)$ in terms of $P(A|B^c)$?

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

Flashcard 32: Calculate $P(A|B)$ for $P(A \cap B) = 0.15$ and $P(B) = 0.3$.

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

Flashcard 33: What is $P(B \cap A)$ if $P(A \cap B) = 0.3$?

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

Flashcard 34: If $P(A|B) = 0.4$, what is $P(A^c|B)$?

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

Flashcard 35: Find $P(A|B)$ with $P(A \cap B) = 0.25$ and $P(B) = 0.5$.

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

Flashcard 36: What is the complementary rule for conditional probability?

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

Flashcard 37: Define $P(A \cap B)$ in terms of conditional probability.

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

Flashcard 38: What is the probability of $A$ given $A$ and $B$ are disjoint?

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

Flashcard 39: State Bayes' Theorem.

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

Flashcard 40: Define mutually exclusive events in terms of $P(A|B)$.

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

Flashcard 41: Find $P(A \cap B)$ for independent events $A$ and $B$. $P(A)=0.4$, $P(B)=0.5$.

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

Flashcard 42: What does $P(A|B) = 1$ imply about the events?

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

Flashcard 43: Determine $P(B|A)$ if $P(A|B)=0.5$, $P(B)=0.4$, $P(A)=0.2$.

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

Flashcard 44: What does $P(B|A) = 1$ suggest?

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

Flashcard 45: What does $P(A|B) = P(A)$ signify about events $A$ and $B$?

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

Flashcard 46: Determine if $P(A|B) = 0.2$ and $P(A) = 0.3$ indicate independence.

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

Flashcard 47: Which theorem is used to find $P(B|A)$ from $P(A|B)$?

Answer: Bayes' Theorem. Used to reverse conditional probability relationships.

Flashcard 48: If $P(A|B)=0.2$, $P(B)=0.6$, find $P(A \cap B)$.

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

Flashcard 49: What is the definition of conditional probability?

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

Flashcard 50: What is $P(A \cap B^c)$ in terms of $P(A|B^c)$?

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

Flashcard 51: What does $P(B|A) = 1$ suggest?

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

Flashcard 52: What does $P(A \cap B) = 0$ indicate?

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

Flashcard 53: Determine if $P(A|B) = 0.2$ and $P(A) = 0.3$ indicate independence.

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

Flashcard 54: State the formula for conditional probability.

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

Flashcard 55: What is $P(B \cap A)$ if $P(A \cap B) = 0.3$?

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

Flashcard 56: Identify $P(B|A)$ if $P(A \cap B) = 0.2$ and $P(A) = 0.4$.

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

Flashcard 57: What is the condition for $P(A|B)=P(A|B^c)$?

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

Flashcard 58: Identify the type of probability: $P(A|B)$.

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

Flashcard 59: State the formula for $P(A^c \cap B)$ using conditional probability.

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

Flashcard 60: Determine if $P(A|B) = 0.3$ and $P(A) = 0.5$ indicate independence.

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

Flashcard 61: What does $P(A|B) = 0$ imply?

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

Flashcard 62: State Bayes' Theorem.

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

Flashcard 63: Calculate $P(A|B)$ if $P(A \cap B) = 0.4$ and $P(B) = 0.8$.

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

Flashcard 64: Evaluate independence: $P(A|B)=0.7$, $P(A)=0.7$.

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

Flashcard 65: Find $P(A|B)$ with $P(A \cap B) = 0.25$ and $P(B) = 0.5$.

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

Flashcard 66: If $P(A|B)=0.6$ and $P(B)=0.5$, find $P(A \cap B)$.

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

Flashcard 67: What does $P(A|B) = P(A)$ signify about events $A$ and $B$?

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

Flashcard 68: What is the condition for two events to be independent?

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

Flashcard 69: For independent events, what is $P(A \cap B)$?

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

Flashcard 70: State the formula for conditional probability.

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

Flashcard 71: Identify $P(A|B)$ given $P(A \cap B) = 0.1$ and $P(B) = 0.25$.

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

Flashcard 72: Which rule relates joint probability to conditional probability?

Answer: Multiplication Rule. Connects joint and conditional probabilities through multiplication.

Flashcard 73: Which theorem is used to find $P(B|A)$ from $P(A|B)$?

Answer: Bayes' Theorem. Used to reverse conditional probability relationships.

Flashcard 74: What is the condition for $P(A|B)=P(A|B^c)$?

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

Flashcard 75: Calculate $P(A|B)$ for $P(A \cap B) = 0.15$ and $P(B) = 0.3$.

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

Flashcard 76: Evaluate independence: $P(A|B)=0.7$, $P(A)=0.7$.

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