Conditional Probability - AP Statistics
Card 1 of 30
What is the condition for two events to be independent?
What is the condition for two events to be independent?
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$P(A|B) = P(A)$ and $P(B|A) = P(B)$. Conditional probability equals marginal probability for independent events.
$P(A|B) = P(A)$ and $P(B|A) = P(B)$. Conditional probability equals marginal probability for independent events.
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Determine if $P(A|B) = 0.3$ and $P(A) = 0.5$ indicate independence.
Determine if $P(A|B) = 0.3$ and $P(A) = 0.5$ indicate independence.
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Not independent. Since $P(A|B) \neq P(A)$, the events are dependent.
Not independent. Since $P(A|B) \neq P(A)$, the events are dependent.
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What does $P(A|B) = 1$ imply about the events?
What does $P(A|B) = 1$ imply about the events?
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Event $A$ occurs whenever $B$ occurs. Event $A$ is certain to happen when $B$ has occurred.
Event $A$ occurs whenever $B$ occurs. Event $A$ is certain to happen when $B$ has occurred.
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Calculate $P(A|B)$ for $P(A \cap B) = 0.15$ and $P(B) = 0.3$.
Calculate $P(A|B)$ for $P(A \cap B) = 0.15$ and $P(B) = 0.3$.
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$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.15}{0.3} = 0.5$.
$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.15}{0.3} = 0.5$.
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What is the complementary rule for conditional probability?
What is the complementary rule for conditional probability?
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$P(A^c|B) = 1 - P(A|B)$. Complement of conditional probability within the same condition.
$P(A^c|B) = 1 - P(A|B)$. Complement of conditional probability within the same condition.
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If $P(A|B) = 0.4$, what is $P(A^c|B)$?
If $P(A|B) = 0.4$, what is $P(A^c|B)$?
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$P(A^c|B) = 0.6$. Since $P(A|B) + P(A^c|B) = 1$.
$P(A^c|B) = 0.6$. Since $P(A|B) + P(A^c|B) = 1$.
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Which theorem is used to find $P(B|A)$ from $P(A|B)$?
Which theorem is used to find $P(B|A)$ from $P(A|B)$?
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Bayes' Theorem. Used to reverse conditional probability relationships.
Bayes' Theorem. Used to reverse conditional probability relationships.
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State Bayes' Theorem.
State Bayes' Theorem.
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$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$. Allows calculation of reverse conditional probability.
$P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A)}$. Allows calculation of reverse conditional probability.
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Determine $P(B|A)$ if $P(A|B)=0.5$, $P(B)=0.4$, $P(A)=0.2$.
Determine $P(B|A)$ if $P(A|B)=0.5$, $P(B)=0.4$, $P(A)=0.2$.
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$P(B|A) = 1$. Using Bayes' Theorem: $\frac{0.5 \times 0.4}{0.2} = 1$.
$P(B|A) = 1$. Using Bayes' Theorem: $\frac{0.5 \times 0.4}{0.2} = 1$.
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What does $P(A|B) = P(A)$ signify about events $A$ and $B$?
What does $P(A|B) = P(A)$ signify about events $A$ and $B$?
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Events $A$ and $B$ are independent. The occurrence of $B$ doesn't affect the probability of $A$.
Events $A$ and $B$ are independent. The occurrence of $B$ doesn't affect the probability of $A$.
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Identify the type of probability: $P(A|B)$.
Identify the type of probability: $P(A|B)$.
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Conditional Probability. Probability of $A$ given that $B$ has occurred.
Conditional Probability. Probability of $A$ given that $B$ has occurred.
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Find $P(A|B)$ with $P(A \cap B) = 0.25$ and $P(B) = 0.5$.
Find $P(A|B)$ with $P(A \cap B) = 0.25$ and $P(B) = 0.5$.
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$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.25}{0.5} = 0.5$.
$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.25}{0.5} = 0.5$.
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What does $P(A|B) = 0$ imply?
What does $P(A|B) = 0$ imply?
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Event $A$ cannot occur if $B$ has occurred. Event $A$ is impossible when $B$ has already happened.
Event $A$ cannot occur if $B$ has occurred. Event $A$ is impossible when $B$ has already happened.
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For independent events, what is $P(A \cap B)$?
For independent events, what is $P(A \cap B)$?
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$P(A \cap B) = P(A) \cdot P(B)$. For independent events, joint probability is the product of marginals.
$P(A \cap B) = P(A) \cdot P(B)$. For independent events, joint probability is the product of marginals.
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Calculate $P(A|B)$ if $P(A \cap B) = 0.4$ and $P(B) = 0.8$.
Calculate $P(A|B)$ if $P(A \cap B) = 0.4$ and $P(B) = 0.8$.
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$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.4}{0.8} = 0.5$.
$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.4}{0.8} = 0.5$.
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Define $P(A \cap B)$ in terms of conditional probability.
Define $P(A \cap B)$ in terms of conditional probability.
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$P(A \cap B) = P(A|B) \cdot P(B)$. Joint probability expressed using conditional probability formula.
$P(A \cap B) = P(A|B) \cdot P(B)$. Joint probability expressed using conditional probability formula.
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What does $P(B|A) = 1$ suggest?
What does $P(B|A) = 1$ suggest?
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Event $B$ occurs whenever $A$ occurs. Event $B$ is certain to occur when $A$ has happened.
Event $B$ occurs whenever $A$ occurs. Event $B$ is certain to occur when $A$ has happened.
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If $P(A|B)=0.6$ and $P(B)=0.5$, find $P(A \cap B)$.
If $P(A|B)=0.6$ and $P(B)=0.5$, find $P(A \cap B)$.
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$P(A \cap B) = 0.3$. Using multiplication rule: $0.6 \times 0.5 = 0.3$.
$P(A \cap B) = 0.3$. Using multiplication rule: $0.6 \times 0.5 = 0.3$.
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State the formula for $P(A^c \cap B)$ using conditional probability.
State the formula for $P(A^c \cap B)$ using conditional probability.
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$P(A^c \cap B) = P(B) - P(A \cap B)$. Probability of complement of $A$ intersecting with $B$.
$P(A^c \cap B) = P(B) - P(A \cap B)$. Probability of complement of $A$ intersecting with $B$.
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Determine $P(A|B)$ if $P(A \cap B) = 0.28$ and $P(B) = 0.7$.
Determine $P(A|B)$ if $P(A \cap B) = 0.28$ and $P(B) = 0.7$.
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$P(A|B) = 0.4$. Using $P(A|B) = \frac{0.28}{0.7} = 0.4$.
$P(A|B) = 0.4$. Using $P(A|B) = \frac{0.28}{0.7} = 0.4$.
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What is $P(B \cap A)$ if $P(A \cap B) = 0.3$?
What is $P(B \cap A)$ if $P(A \cap B) = 0.3$?
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$P(B \cap A) = 0.3$. Intersection is commutative; order doesn't matter.
$P(B \cap A) = 0.3$. Intersection is commutative; order doesn't matter.
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Identify $P(B|A)$ if $P(A \cap B) = 0.2$ and $P(A) = 0.4$.
Identify $P(B|A)$ if $P(A \cap B) = 0.2$ and $P(A) = 0.4$.
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$P(B|A) = 0.5$. Using $P(B|A) = \frac{0.2}{0.4} = 0.5$.
$P(B|A) = 0.5$. Using $P(B|A) = \frac{0.2}{0.4} = 0.5$.
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If $P(A|B)=0.2$, $P(B)=0.6$, find $P(A \cap B)$.
If $P(A|B)=0.2$, $P(B)=0.6$, find $P(A \cap B)$.
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$P(A \cap B) = 0.12$. Using multiplication rule: $0.2 \times 0.6 = 0.12$.
$P(A \cap B) = 0.12$. Using multiplication rule: $0.2 \times 0.6 = 0.12$.
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What is the condition for $P(A|B)=P(A|B^c)$?
What is the condition for $P(A|B)=P(A|B^c)$?
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Events $A$ and $B$ are independent. Knowledge of $B$ or $B^c$ doesn't affect probability of $A$.
Events $A$ and $B$ are independent. Knowledge of $B$ or $B^c$ doesn't affect probability of $A$.
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Identify $P(A|B)$ given $P(A \cap B) = 0.1$ and $P(B) = 0.25$.
Identify $P(A|B)$ given $P(A \cap B) = 0.1$ and $P(B) = 0.25$.
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$P(A|B) = 0.4$. Using $P(A|B) = \frac{0.1}{0.25} = 0.4$.
$P(A|B) = 0.4$. Using $P(A|B) = \frac{0.1}{0.25} = 0.4$.
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What is $P(A \cap B^c)$ in terms of $P(A|B^c)$?
What is $P(A \cap B^c)$ in terms of $P(A|B^c)$?
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$P(A \cap B^c) = P(A|B^c) \cdot P(B^c)$. Joint probability for event $A$ and complement of $B$.
$P(A \cap B^c) = P(A|B^c) \cdot P(B^c)$. Joint probability for event $A$ and complement of $B$.
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Determine if $P(A|B) = 0.2$ and $P(A) = 0.3$ indicate independence.
Determine if $P(A|B) = 0.2$ and $P(A) = 0.3$ indicate independence.
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Not independent. Since $P(A|B) \neq P(A)$, events are dependent.
Not independent. Since $P(A|B) \neq P(A)$, events are dependent.
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What is the definition of conditional probability?
What is the definition of conditional probability?
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Probability of an event given another has occurred. The likelihood of event $A$ happening when we know $B$ has occurred.
Probability of an event given another has occurred. The likelihood of event $A$ happening when we know $B$ has occurred.
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Identify $P(A|B)$ if $P(A \cap B) = 0.3$ and $P(B) = 0.6$.
Identify $P(A|B)$ if $P(A \cap B) = 0.3$ and $P(B) = 0.6$.
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$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.3}{0.6} = 0.5$.
$P(A|B) = 0.5$. Using $P(A|B) = \frac{0.3}{0.6} = 0.5$.
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Which rule relates joint probability to conditional probability?
Which rule relates joint probability to conditional probability?
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Multiplication Rule. Connects joint and conditional probabilities through multiplication.
Multiplication Rule. Connects joint and conditional probabilities through multiplication.
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