Conditional Probability and Independence - Statistics
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Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.60$.
Find $P(A\cap B)$ if $P(A\mid B)=0.25$ and $P(B)=0.60$.
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$0.15$. Multiply: $P(A\cap B)=0.25\times 0.60=0.15$.
$0.15$. Multiply: $P(A\cap B)=0.25\times 0.60=0.15$.
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Find $P(B)$ if $P(A\cap B)=0.09$ and $P(A\mid B)=0.30$.
Find $P(B)$ if $P(A\cap B)=0.09$ and $P(A\mid B)=0.30$.
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$0.3$. Rearrange: $P(B)=\frac{0.09}{0.30}=0.3$.
$0.3$. Rearrange: $P(B)=\frac{0.09}{0.30}=0.3$.
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Decide whether $A$ and $B$ are independent if $P(A)=0.50$ and $P(A\mid B)=0.50$.
Decide whether $A$ and $B$ are independent if $P(A)=0.50$ and $P(A\mid B)=0.50$.
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Independent. Since $P(A\mid B)=P(A)$, they're independent.
Independent. Since $P(A\mid B)=P(A)$, they're independent.
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Decide whether $A$ and $B$ are independent if $P(A)=0.40$ and $P(A\mid B)=0.25$.
Decide whether $A$ and $B$ are independent if $P(A)=0.40$ and $P(A\mid B)=0.25$.
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Not independent. Since $0.25\neq 0.40$, conditioning changes probability.
Not independent. Since $0.25\neq 0.40$, conditioning changes probability.
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Decide whether $A$ and $B$ are independent if $P(A\cap B)=0.10$, $P(A)=0.20$, and $P(B)=0.50$.
Decide whether $A$ and $B$ are independent if $P(A\cap B)=0.10$, $P(A)=0.20$, and $P(B)=0.50$.
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Independent. Check: $0.10=0.20\times 0.50$ confirms independence.
Independent. Check: $0.10=0.20\times 0.50$ confirms independence.
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Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.72$.
Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.72$.
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$P(A\mid B)=0.72$. Independence means $P(A\mid B)=P(A)$.
$P(A\mid B)=0.72$. Independence means $P(A\mid B)=P(A)$.
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Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.15$.
Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.15$.
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$P(B\mid A)=0.15$. Independence means $P(B\mid A)=P(B)$.
$P(B\mid A)=0.15$. Independence means $P(B\mid A)=P(B)$.
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Identify the error: A student writes $P(A\mid B)=\frac{P(A\cup B)}{P(B)}$. What is the correction?
Identify the error: A student writes $P(A\mid B)=\frac{P(A\cup B)}{P(B)}$. What is the correction?
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Use $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Union $\cup$ should be intersection $\cap$.
Use $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Union $\cup$ should be intersection $\cap$.
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Which probability is updated by new information: $P(A)$ or $P(A\mid B)$?
Which probability is updated by new information: $P(A)$ or $P(A\mid B)$?
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$P(A\mid B)$. Conditioning incorporates new information.
$P(A\mid B)$. Conditioning incorporates new information.
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What does the notation $P(A\mid B)$ mean in words?
What does the notation $P(A\mid B)$ mean in words?
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Probability that $A$ occurs given that $B$ has occurred. The vertical bar means "given" or "assuming."
Probability that $A$ occurs given that $B$ has occurred. The vertical bar means "given" or "assuming."
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State the formula for conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$.
State the formula for conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$.
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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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Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.30$, and $P(B)=0.40$.
Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.30$, and $P(B)=0.40$.
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$0.12$. For independent events, multiply probabilities.
$0.12$. For independent events, multiply probabilities.
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State the multiplication rule that expresses $P(A\cap B)$ using $P(A\mid B)$ and $P(B)$.
State the multiplication rule that expresses $P(A\cap B)$ using $P(A\mid B)$ and $P(B)$.
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$P(A\cap B)=P(A\mid B)P(B)$. Rearranges the conditional probability formula.
$P(A\cap B)=P(A\mid B)P(B)$. Rearranges the conditional probability formula.
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State the multiplication rule that expresses $P(A\cap B)$ using $P(B\mid A)$ and $P(A)$.
State the multiplication rule that expresses $P(A\cap B)$ using $P(B\mid A)$ and $P(A)$.
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$P(A\cap B)=P(B\mid A)P(A)$. Alternative form using reversed conditioning.
$P(A\cap B)=P(B\mid A)P(A)$. Alternative form using reversed conditioning.
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What condition must be true for $P(A\mid B)$ to be defined?
What condition must be true for $P(A\mid B)$ to be defined?
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$P(B)>0$. Can't condition on an impossible event.
$P(B)>0$. Can't condition on an impossible event.
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State the independence condition using conditional probability of $A$ given $B$.
State the independence condition using conditional probability of $A$ given $B$.
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$A$ and $B$ are independent if $P(A\mid B)=P(A)$. Knowing $B$ doesn't change $A$'s probability.
$A$ and $B$ are independent if $P(A\mid B)=P(A)$. Knowing $B$ doesn't change $A$'s probability.
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State the independence condition using conditional probability of $B$ given $A$.
State the independence condition using conditional probability of $B$ given $A$.
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$A$ and $B$ are independent if $P(B\mid A)=P(B)$. Knowing $A$ doesn't change $B$'s probability.
$A$ and $B$ are independent if $P(B\mid A)=P(B)$. Knowing $A$ doesn't change $B$'s probability.
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State an equivalent independence condition using only $P(A\cap B)$, $P(A)$, and $P(B)$.
State an equivalent independence condition using only $P(A\cap B)$, $P(A)$, and $P(B)$.
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$A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. Product rule holds when events don't influence each other.
$A$ and $B$ are independent if $P(A\cap B)=P(A)P(B)$. Product rule holds when events don't influence each other.
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Identify the correct expression for $P(A\cap B)$ in terms of $P(A)$ and $P(B\mid A)$.
Identify the correct expression for $P(A\cap B)$ in terms of $P(A)$ and $P(B\mid A)$.
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$P(A\cap B)=P(A)P(B\mid A)$. Applies the multiplication rule correctly.
$P(A\cap B)=P(A)P(B\mid A)$. Applies the multiplication rule correctly.
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Find $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.30$.
Find $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.30$.
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$0.4$. Apply $P(A\mid B)=\frac{0.12}{0.30}=0.4$.
$0.4$. Apply $P(A\mid B)=\frac{0.12}{0.30}=0.4$.
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Decide whether $A$ and $B$ are independent: $P(A)=0.6$, $P(B)=0.5$, $P(A\cap B)=0.25$.
Decide whether $A$ and $B$ are independent: $P(A)=0.6$, $P(B)=0.5$, $P(A\cap B)=0.25$.
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Not independent, since $P(A)P(B)=0.3\ne 0.25$. Checks if $P(A\cap B)=P(A)P(B)$: $0.25\ne 0.6\times 0.5=0.3$ ✗.
Not independent, since $P(A)P(B)=0.3\ne 0.25$. Checks if $P(A\cap B)=P(A)P(B)$: $0.25\ne 0.6\times 0.5=0.3$ ✗.
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Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.80$.
Find $P(B\mid A)$ if $A$ and $B$ are independent and $P(B)=0.80$.
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$P(B\mid A)=0.80$. For independent events, $P(B\mid A)=P(B)$ regardless of $A$.
$P(B\mid A)=0.80$. For independent events, $P(B\mid A)=P(B)$ regardless of $A$.
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Compute $P(A\mid B)$ from the table: $A\cap B=15$, $A^c\cap B=35$.
Compute $P(A\mid B)$ from the table: $A\cap B=15$, $A^c\cap B=35$.
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$P(A\mid B)=\frac{15}{50}=0.3$. Total in $B$ is $15+35=50$, so $P(A\mid B)=\frac{15}{50}$.
$P(A\mid B)=\frac{15}{50}=0.3$. Total in $B$ is $15+35=50$, so $P(A\mid B)=\frac{15}{50}$.
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Identify the value of $P(A\cap B)$ from $P(A)=0.3$, $P(B)=0.5$, assuming independence.
Identify the value of $P(A\cap B)$ from $P(A)=0.3$, $P(B)=0.5$, assuming independence.
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$P(A\cap B)=0.15$. Uses independence formula: $P(A\cap B)=P(A)P(B)=0.3\times 0.5$.
$P(A\cap B)=0.15$. Uses independence formula: $P(A\cap B)=P(A)P(B)=0.3\times 0.5$.
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State the independence rule that connects $P(A\cap B)$, $P(A)$, and $P(B)$.
State the independence rule that connects $P(A\cap B)$, $P(A)$, and $P(B)$.
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If independent, $P(A\cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.
If independent, $P(A\cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.
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Which option is equivalent to independence: $P(A\mid B)=P(A)$ or $P(A\mid B)=P(B)$?
Which option is equivalent to independence: $P(A\mid B)=P(A)$ or $P(A\mid B)=P(B)$?
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$P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability, not that they're equal.
$P(A\mid B)=P(A)$. Independence means conditioning doesn't change probability, not that they're equal.
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Identify the correct relationship when $A$ and $B$ are independent and $P(B)>0$.
Identify the correct relationship when $A$ and $B$ are independent and $P(B)>0$.
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$P(A\cap B)=P(A\mid B)P(B)=P(A)P(B)$. Substitutes $P(A)$ for $P(A\mid B)$ when events are independent.
$P(A\cap B)=P(A\mid B)P(B)=P(A)P(B)$. Substitutes $P(A)$ for $P(A\mid B)$ when events are independent.
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Find $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.30$.
Find $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.30$.
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$P(A\mid B)=0.4$. Uses $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.12}{0.30}$.
$P(A\mid B)=0.4$. Uses $P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.12}{0.30}$.
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Find $P(A)$ assuming independence, given $P(A\cap B)=0.14$ and $P(B)=0.70$.
Find $P(A)$ assuming independence, given $P(A\cap B)=0.14$ and $P(B)=0.70$.
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$P(A)=0.2$. Uses independence: $P(A)=\frac{P(A\cap B)}{P(B)}=\frac{0.14}{0.70}$.
$P(A)=0.2$. Uses independence: $P(A)=\frac{P(A\cap B)}{P(B)}=\frac{0.14}{0.70}$.
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Decide whether $A$ and $B$ are independent: $P(A)=0.5$, $P(B)=0.4$, $P(A\cap B)=0.2$.
Decide whether $A$ and $B$ are independent: $P(A)=0.5$, $P(B)=0.4$, $P(A\cap B)=0.2$.
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Independent, since $P(A\cap B)=P(A)P(B)=0.2$. Checks if $P(A\cap B)=P(A)P(B)$: $0.2=0.5\times 0.4$ ✓.
Independent, since $P(A\cap B)=P(A)P(B)=0.2$. Checks if $P(A\cap B)=P(A)P(B)$: $0.2=0.5\times 0.4$ ✓.
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