Explaining Conditional Probability in Everyday Situations - Statistics
Card 1 of 30
State the multiplication rule that characterizes independence of $A$ and $B$.
State the multiplication rule that characterizes independence of $A$ and $B$.
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Independent if $P(A\cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.
Independent if $P(A\cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.
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Identify the correct complement rule for conditional probability: $P(A^c\mid B)=$ ?
Identify the correct complement rule for conditional probability: $P(A^c\mid B)=$ ?
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$P(A^c\mid B)=1-P(A\mid B)$. Within condition $B$, probabilities of $A$ and $A^c$ must sum to 1.
$P(A^c\mid B)=1-P(A\mid B)$. Within condition $B$, probabilities of $A$ and $A^c$ must sum to 1.
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Compute $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.30$.
Compute $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.30$.
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$0.4$. Apply formula: $P(A\mid B) = \frac{0.12}{0.30} = 0.4$.
$0.4$. Apply formula: $P(A\mid B) = \frac{0.12}{0.30} = 0.4$.
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Identify the condition needed for $P(A\mid B)$ to be defined using $\frac{P(A\cap B)}{P(B)}$.
Identify the condition needed for $P(A\mid B)$ to be defined using $\frac{P(A\cap B)}{P(B)}$.
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$P(B)>0$. Can't divide by zero, so the condition must have positive probability.
$P(B)>0$. Can't divide by zero, so the condition must have positive probability.
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What does it mean in words if $P(A\mid B)>P(A)$?
What does it mean in words if $P(A\mid B)>P(A)$?
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Knowing $B$ increases the chance of $A$. $B$ makes $A$ more likely than without knowing $B$.
Knowing $B$ increases the chance of $A$. $B$ makes $A$ more likely than without knowing $B$.
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What is the symmetry fact about independence: if $A$ is independent of $B$, then what about $B$ and $A$?
What is the symmetry fact about independence: if $A$ is independent of $B$, then what about $B$ and $A$?
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$B$ is independent of $A$. Independence is symmetric - works both ways.
$B$ is independent of $A$. Independence is symmetric - works both ways.
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What is the definition of independence of events $A$ and $B$ using conditional probability?
What is the definition of independence of events $A$ and $B$ using conditional probability?
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$A$ and $B$ are independent if $P(A\mid B)=P(A)$. Knowing $B$ doesn't change the probability of $A$.
$A$ and $B$ are independent if $P(A\mid B)=P(A)$. Knowing $B$ doesn't change the probability of $A$.
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Which probability describes "chance of being a smoker if you have lung cancer": $P(C\mid S)$ or $P(S\mid C)$?
Which probability describes "chance of being a smoker if you have lung cancer": $P(C\mid S)$ or $P(S\mid C)$?
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$P(S\mid C)$. Read as "smoking given cancer" - cancer is the known condition.
$P(S\mid C)$. Read as "smoking given cancer" - cancer is the known condition.
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What does the notation $P(A\cap B)$ mean in everyday language?
What does the notation $P(A\cap B)$ mean in everyday language?
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Probability that both $A$ and $B$ occur. The intersection symbol $\cap$ represents "and" in probability.
Probability that both $A$ and $B$ occur. The intersection symbol $\cap$ represents "and" in probability.
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What does the notation $P(A\cup B)$ mean in everyday language?
What does the notation $P(A\cup B)$ mean in everyday language?
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Probability that $A$ or $B$ (or both) occurs. The union symbol $\cup$ represents "or" in probability.
Probability that $A$ or $B$ (or both) occurs. The union symbol $\cup$ represents "or" in probability.
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Compute $P(A\cap B)$ given $P(A\mid B)=0.25$ and $P(B)=0.60$.
Compute $P(A\cap B)$ given $P(A\mid B)=0.25$ and $P(B)=0.60$.
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$0.15$. Rearrange formula: $P(A\cap B) = P(A\mid B) \times P(B) = 0.25 \times 0.60$.
$0.15$. Rearrange formula: $P(A\cap B) = P(A\mid B) \times P(B) = 0.25 \times 0.60$.
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Decide if $A$ and $B$ are independent when $P(A)=0.50$ and $P(A\mid B)=0.50$.
Decide if $A$ and $B$ are independent when $P(A)=0.50$ and $P(A\mid B)=0.50$.
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Independent. Since $P(A\mid B) = P(A)$, knowing $B$ doesn't affect $A$'s probability.
Independent. Since $P(A\mid B) = P(A)$, knowing $B$ doesn't affect $A$'s probability.
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Decide if $A$ and $B$ are independent when $P(A)=0.50$ and $P(A\mid B)=0.30$.
Decide if $A$ and $B$ are independent when $P(A)=0.50$ and $P(A\mid B)=0.30$.
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Not independent. Since $P(A\mid B) \neq P(A)$, knowing $B$ changes $A$'s probability.
Not independent. Since $P(A\mid B) \neq P(A)$, knowing $B$ changes $A$'s probability.
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Compute $P(B\mid A)$ given $P(A\cap B)=0.18$ and $P(A)=0.60$.
Compute $P(B\mid A)$ given $P(A\cap B)=0.18$ and $P(A)=0.60$.
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$0.3$. Apply formula: $P(B\mid A) = \frac{0.18}{0.60} = 0.3$.
$0.3$. Apply formula: $P(B\mid A) = \frac{0.18}{0.60} = 0.3$.
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What does it mean in words if $P(A\mid B)<P(A)$?
What does it mean in words if $P(A\mid B)<P(A)$?
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Knowing $B$ decreases the chance of $A$. $B$ makes $A$ less likely than without knowing $B$.
Knowing $B$ decreases the chance of $A$. $B$ makes $A$ less likely than without knowing $B$.
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Decide if $A$ and $B$ are independent when $P(A)=0.20$, $P(B)=0.40$, and $P(A\cap B)=0.08$.
Decide if $A$ and $B$ are independent when $P(A)=0.20$, $P(B)=0.40$, and $P(A\cap B)=0.08$.
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Independent. Check: $P(A)P(B) = 0.20 \times 0.40 = 0.08 = P(A\cap B)$.
Independent. Check: $P(A)P(B) = 0.20 \times 0.40 = 0.08 = P(A\cap B)$.
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Which probability describes "chance of lung cancer if you are a smoker": $P(C\mid S)$ or $P(S\mid C)$?
Which probability describes "chance of lung cancer if you are a smoker": $P(C\mid S)$ or $P(S\mid C)$?
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$P(C\mid S)$. Read as "cancer given smoking" - smoking is the known condition.
$P(C\mid S)$. Read as "cancer given smoking" - smoking is the known condition.
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What is the definition of conditional probability $P(A\mid B)$ in words?
What is the definition of conditional probability $P(A\mid B)$ in words?
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Probability of $A$ given that $B$ has occurred. The condition restricts the sample space to outcomes where $B$ occurred.
Probability of $A$ given that $B$ has occurred. The condition restricts the sample space to outcomes where $B$ occurred.
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State the formula for conditional probability using intersection: $P(A\mid B)=$ ?
State the formula for conditional probability using intersection: $P(A\mid B)=$ ?
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$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$. Divides joint probability by the probability of the condition.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ for $P(B)>0$. Divides joint probability by the probability of the condition.
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Which statement is correct: $P(A\mid B)=P(B\mid A)$ always, or they can differ?
Which statement is correct: $P(A\mid B)=P(B\mid A)$ always, or they can differ?
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They can differ. These are different unless $P(A) = P(B)$ and events are independent.
They can differ. These are different unless $P(A) = P(B)$ and events are independent.
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Find $P(A)$ if $A$ and $B$ are independent, $P(A\mid B)=0.30$, and $P(B)>0$.
Find $P(A)$ if $A$ and $B$ are independent, $P(A\mid B)=0.30$, and $P(B)>0$.
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$0.30$. Independence means $P(A\mid B)=P(A)$.
$0.30$. Independence means $P(A\mid B)=P(A)$.
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What is the meaning of conditional probability $P(A\mid B)$ in everyday language?
What is the meaning of conditional probability $P(A\mid B)$ in everyday language?
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$P(A\mid B)$ is the chance $A$ occurs given that $B$ occurred. The vertical bar means "given that" or "assuming".
$P(A\mid B)$ is the chance $A$ occurs given that $B$ occurred. The vertical bar means "given that" or "assuming".
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State the formula for conditional probability $P(A\mid B)$ using $P(A\cap B)$ and $P(B)$.
State the formula for conditional probability $P(A\mid B)$ using $P(A\cap B)$ and $P(B)$.
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$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, for $P(B)>0$. Divides joint probability by the condition's probability.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, for $P(B)>0$. Divides joint probability by the condition's probability.
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What does the notation $A\cap B$ mean in probability language?
What does the notation $A\cap B$ mean in probability language?
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$A\cap B$ means both events $A$ and $B$ happen. The intersection symbol represents simultaneous occurrence.
$A\cap B$ means both events $A$ and $B$ happen. The intersection symbol represents simultaneous occurrence.
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What does the notation $A^c$ mean in probability language?
What does the notation $A^c$ mean in probability language?
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$A^c$ means event $A$ does not happen (the complement of $A$). The superscript $c$ denotes "complement" or "not".
$A^c$ means event $A$ does not happen (the complement of $A$). The superscript $c$ denotes "complement" or "not".
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What is the meaning of independence for events $A$ and $B$ in everyday language?
What is the meaning of independence for events $A$ and $B$ in everyday language?
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Knowing $B$ happened does not change the chance that $A$ happens. Independence means one event doesn't affect the other's probability.
Knowing $B$ happened does not change the chance that $A$ happens. Independence means one event doesn't affect the other's probability.
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State an equation that shows $A$ and $B$ are independent using conditional probability.
State an equation that shows $A$ and $B$ are independent using conditional probability.
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$P(A\mid B)=P(A)$, with $P(B)>0$. For independent events, conditioning doesn't change probability.
$P(A\mid B)=P(A)$, with $P(B)>0$. For independent events, conditioning doesn't change probability.
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State an equation that shows $A$ and $B$ are independent using intersection probability.
State an equation that shows $A$ and $B$ are independent using intersection probability.
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$P(A\cap B)=P(A),P(B)$. Independent events multiply to get joint probability.
$P(A\cap B)=P(A),P(B)$. Independent events multiply to get joint probability.
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Identify the key difference between $P(A\mid B)$ and $P(B\mid A)$.
Identify the key difference between $P(A\mid B)$ and $P(B\mid A)$.
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They reverse the condition; they are generally not equal. Order matters: what's given vs. what we're finding.
They reverse the condition; they are generally not equal. Order matters: what's given vs. what we're finding.
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Which probability matches the phrase "chance of lung cancer if you are a smoker"?
Which probability matches the phrase "chance of lung cancer if you are a smoker"?
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$P(\text{cancer}\mid\text{smoker})$. "If" indicates the condition (what's given).
$P(\text{cancer}\mid\text{smoker})$. "If" indicates the condition (what's given).
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