Independent Events and Unions of Events - AP Statistics
Card 1 of 30
What does it mean if $P(A \text{ and } B) \neq P(A) \times P(B)$?
What does it mean if $P(A \text{ and } B) \neq P(A) \times P(B)$?
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Events $A$ and $B$ are not independent. The product rule for independence fails.
Events $A$ and $B$ are not independent. The product rule for independence fails.
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What is the definition of independent events?
What is the definition of independent events?
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Events where $P(A \text{ and } B) = P(A) \times P(B)$. The occurrence of one doesn't affect the other's probability.
Events where $P(A \text{ and } B) = P(A) \times P(B)$. The occurrence of one doesn't affect the other's probability.
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State the formula for $P(A \text{ or } B)$ for any events $A$ and $B$.
State the formula for $P(A \text{ or } B)$ for any events $A$ and $B$.
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$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General addition rule accounting for overlap between events.
$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General addition rule accounting for overlap between events.
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Are events $A$ and $B$ independent if $P(A) = 0.3$, $P(B) = 0.4$, $P(A \text{ and } B) = 0.12$?
Are events $A$ and $B$ independent if $P(A) = 0.3$, $P(B) = 0.4$, $P(A \text{ and } B) = 0.12$?
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Yes, because $0.3 \times 0.4 = 0.12$. The product equals the intersection probability.
Yes, because $0.3 \times 0.4 = 0.12$. The product equals the intersection probability.
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What is the probability of $A \text{ and } B$ for independent events?
What is the probability of $A \text{ and } B$ for independent events?
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$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.
$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.
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Find $P(A \text{ or } B)$ given $P(A) = 0.5$, $P(B) = 0.3$, $P(A \text{ and } B) = 0.1$.
Find $P(A \text{ or } B)$ given $P(A) = 0.5$, $P(B) = 0.3$, $P(A \text{ and } B) = 0.1$.
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$P(A \text{ or } B) = 0.5 + 0.3 - 0.1 = 0.7$. Apply the addition rule: sum minus overlap.
$P(A \text{ or } B) = 0.5 + 0.3 - 0.1 = 0.7$. Apply the addition rule: sum minus overlap.
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State the complement rule for event $A$.
State the complement rule for event $A$.
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$P(A^c) = 1 - P(A)$. Probability of an event plus its complement equals 1.
$P(A^c) = 1 - P(A)$. Probability of an event plus its complement equals 1.
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How do you verify if two events are independent?
How do you verify if two events are independent?
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Check if $P(A \text{ and } B) = P(A) \times P(B)$. Apply the independence condition test.
Check if $P(A \text{ and } B) = P(A) \times P(B)$. Apply the independence condition test.
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What is $P(A^c \text{ or } B)$ if $A$ and $B$ are independent?
What is $P(A^c \text{ or } B)$ if $A$ and $B$ are independent?
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$P(A^c \text{ or } B) = 1 - P(A) \times (1 - P(B))$. Use De Morgan's law and independence property.
$P(A^c \text{ or } B) = 1 - P(A) \times (1 - P(B))$. Use De Morgan's law and independence property.
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If $P(A) = 0.2$ and $P(B) = 0.4$, calculate $P(A \text{ and } B)$ for independent events.
If $P(A) = 0.2$ and $P(B) = 0.4$, calculate $P(A \text{ and } B)$ for independent events.
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$P(A \text{ and } B) = 0.2 \times 0.4 = 0.08$. Multiply the individual probabilities for independence.
$P(A \text{ and } B) = 0.2 \times 0.4 = 0.08$. Multiply the individual probabilities for independence.
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Define a union of two events $A$ and $B$.
Define a union of two events $A$ and $B$.
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The set of outcomes in $A$, $B$, or both. All outcomes where at least one event occurs.
The set of outcomes in $A$, $B$, or both. All outcomes where at least one event occurs.
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What is the probability of the union of $A$ and $B$ if they are mutually exclusive?
What is the probability of the union of $A$ and $B$ if they are mutually exclusive?
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$P(A \text{ or } B) = P(A) + P(B)$. No overlap means no subtraction needed.
$P(A \text{ or } B) = P(A) + P(B)$. No overlap means no subtraction needed.
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Find $P(A^c)$ given $P(A) = 0.75$.
Find $P(A^c)$ given $P(A) = 0.75$.
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$P(A^c) = 1 - 0.75 = 0.25$. Apply the complement rule directly.
$P(A^c) = 1 - 0.75 = 0.25$. Apply the complement rule directly.
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What is the intersection of events $A$ and $B$?
What is the intersection of events $A$ and $B$?
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The set of outcomes in both $A$ and $B$. Outcomes common to both events.
The set of outcomes in both $A$ and $B$. Outcomes common to both events.
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For events $A$ and $B$, state the addition rule for probability.
For events $A$ and $B$, state the addition rule for probability.
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$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. The general formula for union probability.
$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. The general formula for union probability.
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What is $P(A \text{ or } A^c)$?
What is $P(A \text{ or } A^c)$?
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$P(A \text{ or } A^c) = 1$. An event and its complement cover all possibilities.
$P(A \text{ or } A^c) = 1$. An event and its complement cover all possibilities.
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If events $A$ and $B$ are independent, what is $P(A \text{ and } B^c)$?
If events $A$ and $B$ are independent, what is $P(A \text{ and } B^c)$?
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$P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Event $A$ occurs while $B$ doesn't.
$P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Event $A$ occurs while $B$ doesn't.
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If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \text{ or } B) = 0.7$, are $A$ and $B$ independent?
If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \text{ or } B) = 0.7$, are $A$ and $B$ independent?
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No, because $0.4 \times 0.5 \neq 0.2$. Check if $P(A \text{ and } B) = P(A) \times P(B) = 0.2$.
No, because $0.4 \times 0.5 \neq 0.2$. Check if $P(A \text{ and } B) = P(A) \times P(B) = 0.2$.
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For independent events, how to find $P(A \text{ and } B^c)$?
For independent events, how to find $P(A \text{ and } B^c)$?
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$P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Multiply probability of $A$ by complement of $B$.
$P(A \text{ and } B^c) = P(A) \times (1 - P(B))$. Multiply probability of $A$ by complement of $B$.
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Define mutually exclusive events.
Define mutually exclusive events.
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Events that cannot occur simultaneously. Their intersection has zero probability.
Events that cannot occur simultaneously. Their intersection has zero probability.
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Find $P(A \text{ or } B^c)$ given $P(A) = 0.3$, $P(B) = 0.4$, independent events.
Find $P(A \text{ or } B^c)$ given $P(A) = 0.3$, $P(B) = 0.4$, independent events.
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$P(A \text{ or } B^c) = 1 - 0.3 \times 0.4 = 0.88$. Use complement rule: $1 - P(A^c \text{ and } B)$.
$P(A \text{ or } B^c) = 1 - 0.3 \times 0.4 = 0.88$. Use complement rule: $1 - P(A^c \text{ and } B)$.
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What is the probability of $A$ if $P(A \text{ and } B) = 0.2$ and $P(B) = 0.5$, for independent events?
What is the probability of $A$ if $P(A \text{ and } B) = 0.2$ and $P(B) = 0.5$, for independent events?
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$P(A) = \frac{0.2}{0.5} = 0.4$. Divide intersection by the other event's probability.
$P(A) = \frac{0.2}{0.5} = 0.4$. Divide intersection by the other event's probability.
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What does $P(A \text{ and } A^c)$ equal?
What does $P(A \text{ and } A^c)$ equal?
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$P(A \text{ and } A^c) = 0$. An event cannot occur with its complement.
$P(A \text{ and } A^c) = 0$. An event cannot occur with its complement.
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Which rule applies to independent events for conditional probability?
Which rule applies to independent events for conditional probability?
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$P(B|A) = P(B)$. Knowing $A$ doesn't change probability of $B$.
$P(B|A) = P(B)$. Knowing $A$ doesn't change probability of $B$.
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If $P(A) = 0.6$, $P(B \text{ and } A) = 0.24$, find $P(B)$ if independent.
If $P(A) = 0.6$, $P(B \text{ and } A) = 0.24$, find $P(B)$ if independent.
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$P(B) = \frac{0.24}{0.6} = 0.4$. Use independence: divide intersection by $P(A)$.
$P(B) = \frac{0.24}{0.6} = 0.4$. Use independence: divide intersection by $P(A)$.
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What is $P(A^c \text{ or } B^c)$ if $A$ and $B$ are independent?
What is $P(A^c \text{ or } B^c)$ if $A$ and $B$ are independent?
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$P(A^c \text{ or } B^c) = 1 - P(A)P(B)$. Use De Morgan's law: complement of intersection.
$P(A^c \text{ or } B^c) = 1 - P(A)P(B)$. Use De Morgan's law: complement of intersection.
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Find $P(A \text{ or } B)$ if $P(A) = 0.4$, $P(B) = 0.5$, $P(A \text{ and } B) = 0.2$.
Find $P(A \text{ or } B)$ if $P(A) = 0.4$, $P(B) = 0.5$, $P(A \text{ and } B) = 0.2$.
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$P(A \text{ or } B) = 0.4 + 0.5 - 0.2 = 0.7$. Standard addition rule application.
$P(A \text{ or } B) = 0.4 + 0.5 - 0.2 = 0.7$. Standard addition rule application.
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What is the probability of a sure event?
What is the probability of a sure event?
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$P(\text{sure event}) = 1$. Certain events have maximum probability.
$P(\text{sure event}) = 1$. Certain events have maximum probability.
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If $P(A) = 0.3$ and $P(B) = 0.5$, find $P(A \text{ and } B^c)$ if independent.
If $P(A) = 0.3$ and $P(B) = 0.5$, find $P(A \text{ and } B^c)$ if independent.
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$P(A \text{ and } B^c) = 0.3 \times 0.5 = 0.15$. Apply independence rule with complement.
$P(A \text{ and } B^c) = 0.3 \times 0.5 = 0.15$. Apply independence rule with complement.
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What is the probability of $A^c$ and $A$ occurring together?
What is the probability of $A^c$ and $A$ occurring together?
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$P(A^c \text{ and } A) = 0$. Complements are mutually exclusive.
$P(A^c \text{ and } A) = 0$. Complements are mutually exclusive.
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