Finding Conditional Probability in Models - Statistics
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What condition must hold for $P(A\mid B)$ to be defined?
What condition must hold for $P(A\mid B)$ to be defined?
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$P(B)>0$. Cannot divide by zero probability.
$P(B)>0$. Cannot divide by zero probability.
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State the fraction interpretation of $P(A\mid B)$ using counts $|A\cap B|$ and $|B|$.
State the fraction interpretation of $P(A\mid B)$ using counts $|A\cap B|$ and $|B|$.
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$P(A\mid B)=\frac{|A\cap B|}{|B|}$ (equally likely outcomes). Counts favorable outcomes within the restricted set $B$.
$P(A\mid B)=\frac{|A\cap B|}{|B|}$ (equally likely outcomes). Counts favorable outcomes within the restricted set $B$.
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Find $P(A\mid B)$ if $|A\cap B|=18$ and $|B|=60$ in an equally likely model.
Find $P(A\mid B)$ if $|A\cap B|=18$ and $|B|=60$ in an equally likely model.
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$0.30$. Divide favorable count by total in $B$: $\frac{18}{60}=0.30$.
$0.30$. Divide favorable count by total in $B$: $\frac{18}{60}=0.30$.
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Compute $P(A\mid B)$ from a table where $|A\cap B|=25$ and $|B|=100$.
Compute $P(A\mid B)$ from a table where $|A\cap B|=25$ and $|B|=100$.
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$0.25$. Fraction of $B$'s outcomes in $A$: $\frac{25}{100}=0.25$.
$0.25$. Fraction of $B$'s outcomes in $A$: $\frac{25}{100}=0.25$.
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What is $P(A\mid B)$ if $A$ and $B$ are disjoint and $P(B)>0$?
What is $P(A\mid B)$ if $A$ and $B$ are disjoint and $P(B)>0$?
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$0$. Disjoint means $A\cap B=\emptyset$, so $P(A\cap B)=0$.
$0$. Disjoint means $A\cap B=\emptyset$, so $P(A\cap B)=0$.
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State the formula for conditional probability $P(A\mid B)$ using intersection and $P(B)$.
State the formula for conditional probability $P(A\mid B)$ using intersection 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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State the multiplication rule that rewrites $P(A\cap B)$ using $P(A\mid B)$.
State the multiplication rule that rewrites $P(A\cap B)$ using $P(A\mid B)$.
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$P(A\cap B)=P(A\mid B)P(B)$. Rearranges conditional probability formula.
$P(A\cap B)=P(A\mid B)P(B)$. Rearranges conditional probability formula.
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What is $P(A\mid B)$ if $B\subseteq A$ and $P(B)>0$?
What is $P(A\mid B)$ if $B\subseteq A$ and $P(B)>0$?
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$1$. All of $B$ is in $A$, so $P(A\cap B)=P(B)$.
$1$. All of $B$ is in $A$, so $P(A\cap B)=P(B)$.
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What does $P(A\mid B)$ mean in words in terms of outcomes in $B$?
What does $P(A\mid B)$ mean in words in terms of outcomes in $B$?
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Probability of $A$ among outcomes restricted to $B$. Restricts sample space to only outcomes in $B$.
Probability of $A$ among outcomes restricted to $B$. Restricts sample space to only outcomes in $B$.
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Identify the condition required for $P(A\mid B)$ to be defined.
Identify the condition required for $P(A\mid B)$ to be defined.
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$P(B)>0$. Cannot divide by zero probability.
$P(B)>0$. Cannot divide by zero probability.
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Choose the correct denominator for $P(A\mid B)$ in a two-way table: $|A\cap B|$, $|A|$, or $|B|$?
Choose the correct denominator for $P(A\mid B)$ in a two-way table: $|A\cap B|$, $|A|$, or $|B|$?
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Denominator is $|B|$. Restricts to outcomes in condition $B$.
Denominator is $|B|$. Restricts to outcomes in condition $B$.
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Find and correct the error: $P(A\mid B)=\frac{P(A)}{P(B)}$.
Find and correct the error: $P(A\mid B)=\frac{P(A)}{P(B)}$.
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Correct: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Must use intersection, not just $P(A)$.
Correct: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Must use intersection, not just $P(A)$.
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Identify the correct interpretation: If $P(A\mid B)=0.70$, what does $0.70$ describe?
Identify the correct interpretation: If $P(A\mid B)=0.70$, what does $0.70$ describe?
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$70%$ of outcomes in $B$ are also in $A$. Fraction of $B$'s outcomes that belong to $A$.
$70%$ of outcomes in $B$ are also in $A$. Fraction of $B$'s outcomes that belong to $A$.
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A bag has $10$ red and $6$ blue marbles; $4$ red are large and $3$ blue are large. Find $P(\text{red}\mid \text{large})$.
A bag has $10$ red and $6$ blue marbles; $4$ red are large and $3$ blue are large. Find $P(\text{red}\mid \text{large})$.
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$\frac{4}{7}$. Among $7$ large marbles, $4$ are red.
$\frac{4}{7}$. Among $7$ large marbles, $4$ are red.
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In a class, $12$ students are left-handed; $5$ are left-handed and wear glasses. Find $P(\text{glasses}\mid \text{left})$.
In a class, $12$ students are left-handed; $5$ are left-handed and wear glasses. Find $P(\text{glasses}\mid \text{left})$.
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$\frac{5}{12}$. Among $12$ left-handed, $5$ wear glasses.
$\frac{5}{12}$. Among $12$ left-handed, $5$ wear glasses.
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Compute $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.
Compute $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.35$.
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$0.35$. Independence implies $P(A\mid B)=P(A)$.
$0.35$. Independence implies $P(A\mid B)=P(A)$.
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Identify the correct equality for independent events using conditional probability.
Identify the correct equality for independent events using conditional probability.
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If independent, then $P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability.
If independent, then $P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability.
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Find $P(B)$ if $P(A\cap B)=0.18$ and $P(A\mid B)=0.60$.
Find $P(B)$ if $P(A\cap B)=0.18$ and $P(A\mid B)=0.60$.
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$0.30$. Rearrange: $P(B)=\frac{0.18}{0.60}=0.30$.
$0.30$. Rearrange: $P(B)=\frac{0.18}{0.60}=0.30$.
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Find $P(A\cap B)$ if $P(A\mid B)=0.20$ and $P(B)=0.50$.
Find $P(A\cap B)$ if $P(A\mid B)=0.20$ and $P(B)=0.50$.
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$0.10$. Multiply: $0.20 \times 0.50 = 0.10$.
$0.10$. Multiply: $0.20 \times 0.50 = 0.10$.
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State the multiplication rule that rewrites $P(A\cap B)$ using $P(B\mid A)$.
State the multiplication rule that rewrites $P(A\cap B)$ using $P(B\mid A)$.
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$P(A\cap B)=P(B\mid A)P(A)$. Same rule with roles of $A$ and $B$ reversed.
$P(A\cap B)=P(B\mid A)P(A)$. Same rule with roles of $A$ and $B$ reversed.
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If events $A$ and $B$ are disjoint and $P(B)>0$, what is $P(A\mid B)$?
If events $A$ and $B$ are disjoint and $P(B)>0$, what is $P(A\mid B)$?
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$0$. Disjoint events have empty intersection.
$0$. Disjoint events have empty intersection.
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What is the definition of conditional probability $P(A\mid B)$ in terms of outcomes?
What is the definition of conditional probability $P(A\mid B)$ in terms of outcomes?
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$P(A\mid B)=\frac{#(A\cap B)}{#(B)}$ for equally likely outcomes. Counts favorable outcomes in $B$ that also satisfy $A$.
$P(A\mid B)=\frac{#(A\cap B)}{#(B)}$ for equally likely outcomes. Counts favorable outcomes in $B$ that also satisfy $A$.
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State the formula for $P(A\mid B)$ using probabilities (not outcome counts).
State the formula for $P(A\mid B)$ using probabilities (not outcome counts).
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$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, with $P(B)>0$. Divides joint probability by the condition's probability.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, with $P(B)>0$. Divides joint probability by the condition's probability.
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What does $P(A\mid B)$ mean in words in a probability model?
What does $P(A\mid B)$ mean in words in a probability model?
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Probability that $A$ occurs given that $B$ has occurred. Updates probability based on knowing $B$ happened.
Probability that $A$ occurs given that $B$ has occurred. Updates probability based on knowing $B$ happened.
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Identify the sample space used when computing $P(A\mid B)$ from equally likely outcomes.
Identify the sample space used when computing $P(A\mid B)$ from equally likely outcomes.
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The restricted sample space consisting only of outcomes in $B$. Conditioning restricts to outcomes where $B$ occurs.
The restricted sample space consisting only of outcomes in $B$. Conditioning restricts to outcomes where $B$ occurs.
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What is the relationship between $P(A\cap B)$ and $P(A\mid B)$?
What is the relationship between $P(A\cap B)$ and $P(A\mid B)$?
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$P(A\cap B)=P(A\mid B),P(B)$. Multiplication rule for joint probabilities.
$P(A\cap B)=P(A\mid B),P(B)$. Multiplication rule for joint probabilities.
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What is $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?
What is $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?
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$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$, with $P(A)>0$. Swaps roles of $A$ and $B$ in conditional formula.
$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$, with $P(A)>0$. Swaps roles of $A$ and $B$ in conditional formula.
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Which expression equals $P(A\mid B)$: $\frac{P(A\cap B)}{P(B)}$ or $\frac{P(A\cup B)}{P(B)}$?
Which expression equals $P(A\mid B)$: $\frac{P(A\cap B)}{P(B)}$ or $\frac{P(A\cup B)}{P(B)}$?
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$\frac{P(A\cap B)}{P(B)}$. Uses intersection, not union, in numerator.
$\frac{P(A\cap B)}{P(B)}$. Uses intersection, not union, in numerator.
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Find $P(A\mid B)$ if $#(A\cap B)=12$ and $#(B)=30$ (equally likely outcomes).
Find $P(A\mid B)$ if $#(A\cap B)=12$ and $#(B)=30$ (equally likely outcomes).
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$\frac{12}{30}=\frac{2}{5}$. Direct application of outcome-based formula.
$\frac{12}{30}=\frac{2}{5}$. Direct application of outcome-based formula.
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Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.40$.
Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.40$.
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$0.25\times 0.40=0.10$. Uses multiplication rule $P(A\cap B)=P(A\mid B)P(B)$.
$0.25\times 0.40=0.10$. Uses multiplication rule $P(A\cap B)=P(A\mid B)P(B)$.
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