Understanding Independent Events - Statistics
Card 1 of 30
Identify whether $A$ and $B$ are independent from this table: $P(A)=0.4$, $P(B)=0.3$, $P(A\cap B)=0.10$.
Identify whether $A$ and $B$ are independent from this table: $P(A)=0.4$, $P(B)=0.3$, $P(A\cap B)=0.10$.
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Not independent. $0.4 \times 0.3 = 0.12 \neq 0.10$.
Not independent. $0.4 \times 0.3 = 0.12 \neq 0.10$.
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Which formula gives conditional probability $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?
Which formula gives conditional probability $P(B\mid A)$ in terms of $P(A\cap B)$ and $P(A)$?
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$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$. Conditional probability is the ratio of joint to marginal probability.
$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$. Conditional probability is the ratio of joint to marginal probability.
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Which statement is true if $A$ and $B$ are independent: $P(A\cap B)$ equals what expression?
Which statement is true if $A$ and $B$ are independent: $P(A\cap B)$ equals what expression?
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$P(A\cap B)=P(A)P(B)$. This is the defining property of independent events.
$P(A\cap B)=P(A)P(B)$. This is the defining property of independent events.
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Which statement is always true about independence: does $A$ being independent of $B$ imply $B$ independent of $A$?
Which statement is always true about independence: does $A$ being independent of $B$ imply $B$ independent of $A$?
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Yes, independence is symmetric. If $P(A\cap B)=P(A)P(B)$, then both directions hold.
Yes, independence is symmetric. If $P(A\cap B)=P(A)P(B)$, then both directions hold.
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Identify whether $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
Identify whether $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
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Independent. $0.4 \times 0.5 = 0.2$, which equals $P(A\cap B)$.
Independent. $0.4 \times 0.5 = 0.2$, which equals $P(A\cap B)$.
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Find and correct the mistake: claiming independence because $P(A\cap B)=P(A)+P(B)$.
Find and correct the mistake: claiming independence because $P(A\cap B)=P(A)+P(B)$.
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Correct test: $P(A\cap B)=P(A)P(B)$. Addition rule is for disjoint events, not independence.
Correct test: $P(A\cap B)=P(A)P(B)$. Addition rule is for disjoint events, not independence.
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Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.7$ with $P(B)>0$.
Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.7$ with $P(B)>0$.
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Not independent. $P(A\mid B) \neq P(A)$ means not independent.
Not independent. $P(A\mid B) \neq P(A)$ means not independent.
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Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.5$ with $P(B)>0$.
Identify whether $A$ and $B$ are independent if $P(A)=0.5$ and $P(A\mid B)=0.5$ with $P(B)>0$.
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Independent. $P(A\mid B) = P(A)$ confirms independence.
Independent. $P(A\mid B) = P(A)$ confirms independence.
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Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.65$.
Find $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)=0.65$.
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$0.65$. For independent events, $P(A\mid B) = P(A)$.
$0.65$. For independent events, $P(A\mid B) = P(A)$.
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Identify whether $A$ and $B$ are independent if $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{6}$.
Identify whether $A$ and $B$ are independent if $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{6}$.
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Independent. $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$, which equals $P(A\cap B)$.
Independent. $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$, which equals $P(A\cap B)$.
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Find $P(A)$ if $A$ and $B$ are independent, $P(B)=0.8$, and $P(A\cap B)=0.12$.
Find $P(A)$ if $A$ and $B$ are independent, $P(B)=0.8$, and $P(A\cap B)=0.12$.
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$0.15$. Solve $P(A) \times 0.8 = 0.12$ to get $P(A) = 0.15$.
$0.15$. Solve $P(A) \times 0.8 = 0.12$ to get $P(A) = 0.15$.
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Identify whether $A$ and $B$ are independent if $P(A)=0.3$, $P(B)=0.6$, and $P(A\cap B)=0.25$.
Identify whether $A$ and $B$ are independent if $P(A)=0.3$, $P(B)=0.6$, and $P(A\cap B)=0.25$.
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Not independent. $0.3 \times 0.6 = 0.18 \neq 0.25$.
Not independent. $0.3 \times 0.6 = 0.18 \neq 0.25$.
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Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.7$, and $P(B)=0.2$.
Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.7$, and $P(B)=0.2$.
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$0.14$. For independent events, multiply: $0.7 \times 0.2 = 0.14$.
$0.14$. For independent events, multiply: $0.7 \times 0.2 = 0.14$.
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Find $P(B)$ if $A$ and $B$ are independent, $P(A)=0.25$, and $P(A\cap B)=0.05$.
Find $P(B)$ if $A$ and $B$ are independent, $P(A)=0.25$, and $P(A\cap B)=0.05$.
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$0.20$. Solve $0.25 \times P(B) = 0.05$ to get $P(B) = 0.20$.
$0.20$. Solve $0.25 \times P(B) = 0.05$ to get $P(B) = 0.20$.
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What is the independence condition using the product rule for events $A$ and $B$?
What is the independence condition using the product rule for events $A$ and $B$?
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$P(A \cap B)=P(A)P(B)$. Independent events satisfy this multiplication rule.
$P(A \cap B)=P(A)P(B)$. Independent events satisfy this multiplication rule.
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What is the symmetric conditional form of independence between events $A$ and $B$?
What is the symmetric conditional form of independence between events $A$ and $B$?
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$P(A\mid B)=P(A)$ and $P(B\mid A)=P(B)$. Independence means each event's probability is unchanged by the other.
$P(A\mid B)=P(A)$ and $P(B\mid A)=P(B)$. Independence means each event's probability is unchanged by the other.
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What equation tests independence using $P(A\mid B)$ when $P(B)>0$?
What equation tests independence using $P(A\mid B)$ when $P(B)>0$?
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$P(A\mid B)=P(A)$. If $B$ doesn't affect $A$'s probability, they're independent.
$P(A\mid B)=P(A)$. If $B$ doesn't affect $A$'s probability, they're independent.
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What equation tests independence using $P(B\mid A)$ when $P(A)>0$?
What equation tests independence using $P(B\mid A)$ when $P(A)>0$?
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$P(B\mid A)=P(B)$. If $A$ doesn't affect $B$'s probability, they're independent.
$P(B\mid A)=P(B)$. If $A$ doesn't affect $B$'s probability, they're independent.
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Which formula gives conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$?
Which formula gives conditional probability $P(A\mid B)$ in terms of $P(A\cap B)$ and $P(B)$?
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$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability is the ratio of joint to marginal probability.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability is the ratio of joint to marginal probability.
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Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.7$ and $P(B)=0.2$.
Find $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.7$ and $P(B)=0.2$.
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$0.14$. Multiply: $0.7 \times 0.2 = 0.14$.
$0.14$. Multiply: $0.7 \times 0.2 = 0.14$.
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Identify whether $A$ and $B$ are independent if $P(A)=0.6$, $P(B)=0.3$, and $P(A\cap B)=0.25$.
Identify whether $A$ and $B$ are independent if $P(A)=0.6$, $P(B)=0.3$, and $P(A\cap B)=0.25$.
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Not independent. Check: $0.6 \times 0.3 = 0.18 \neq 0.25$.
Not independent. Check: $0.6 \times 0.3 = 0.18 \neq 0.25$.
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What is $P(A \cap B)$ if $A$ and $B$ are independent and you know $P(A)$ and $P(B)$?
What is $P(A \cap B)$ if $A$ and $B$ are independent and you know $P(A)$ and $P(B)$?
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$P(A \cap B)=P(A)P(B)$. For independent events, multiply their individual probabilities.
$P(A \cap B)=P(A)P(B)$. For independent events, multiply their individual probabilities.
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Which probability must equal $P(A)P(B)$ for events $A$ and $B$ to be independent?
Which probability must equal $P(A)P(B)$ for events $A$ and $B$ to be independent?
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$P(A \cap B)$. The intersection probability must equal the product for independence.
$P(A \cap B)$. The intersection probability must equal the product for independence.
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What is the independence test written using conditional probability $P(B\mid A)$?
What is the independence test written using conditional probability $P(B\mid A)$?
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$P(B\mid A)=P(B)$. For independent events, knowing A occurred doesn't change the probability of B.
$P(B\mid A)=P(B)$. For independent events, knowing A occurred doesn't change the probability of B.
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What is the independence test written using conditional probability $P(A\mid B)$?
What is the independence test written using conditional probability $P(A\mid B)$?
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$P(A\mid B)=P(A)$. For independent events, knowing B occurred doesn't change the probability of A.
$P(A\mid B)=P(A)$. For independent events, knowing B occurred doesn't change the probability of A.
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What is the defining equation for independence using the intersection probability?
What is the defining equation for independence using the intersection probability?
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$P(A \cap B)=P(A)P(B)$. Events are independent when their joint probability equals the product of individual probabilities.
$P(A \cap B)=P(A)P(B)$. Events are independent when their joint probability equals the product of individual probabilities.
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Find $P(A)$ if $A$ and $B$ are independent with $P(B)=0.8$ and $P(A\cap B)=0.24$.
Find $P(A)$ if $A$ and $B$ are independent with $P(B)=0.8$ and $P(A\cap B)=0.24$.
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$0.3$. Solve: $P(A) \times 0.8 = 0.24$, so $P(A) = 0.3$.
$0.3$. Solve: $P(A) \times 0.8 = 0.24$, so $P(A) = 0.3$.
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Find $P(B)$ if $A$ and $B$ are independent with $P(A)=0.25$ and $P(A\cap B)=0.05$.
Find $P(B)$ if $A$ and $B$ are independent with $P(A)=0.25$ and $P(A\cap B)=0.05$.
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$0.2$. Solve: $0.25 \times P(B) = 0.05$, so $P(B) = 0.2$.
$0.2$. Solve: $0.25 \times P(B) = 0.05$, so $P(B) = 0.2$.
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Which statement correctly describes independence: $P(A\cap B)>P(A)P(B)$, $=$, or $<$?
Which statement correctly describes independence: $P(A\cap B)>P(A)P(B)$, $=$, or $<$?
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$P(A\cap B)=P(A)P(B)$. Independence requires equality, not greater than or less than.
$P(A\cap B)=P(A)P(B)$. Independence requires equality, not greater than or less than.
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Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{4}$.
Identify whether $A$ and $B$ are independent if $P(A)=\frac{2}{5}$, $P(B)=\frac{1}{2}$, and $P(A\cap B)=\frac{1}{4}$.
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Not independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5} \neq \frac{1}{4}$.
Not independent. Check: $\frac{2}{5} \times \frac{1}{2} = \frac{1}{5} \neq \frac{1}{4}$.
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