Probability - GRE Quantitative Reasoning
Card 1 of 24
What is the addition rule for mutually exclusive events $A$ and $B$?
What is the addition rule for mutually exclusive events $A$ and $B$?
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$P(A\cup B)=P(A)+P(B)$. For mutually exclusive events, there is no overlap, so their union probability is simply the sum of their individual probabilities.
$P(A\cup B)=P(A)+P(B)$. For mutually exclusive events, there is no overlap, so their union probability is simply the sum of their individual probabilities.
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What is the addition rule for two events $A$ and $B$ (not necessarily disjoint)?
What is the addition rule for two events $A$ and $B$ (not necessarily disjoint)?
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$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The addition rule accounts for the overlap by subtracting the intersection probability from the sum of individual probabilities.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The addition rule accounts for the overlap by subtracting the intersection probability from the sum of individual probabilities.
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What is the complement rule for an event $A$?
What is the complement rule for an event $A$?
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$P(A^c)=1-P(A)$. The complement rule states that the probability of an event not occurring equals one minus the probability of it occurring.
$P(A^c)=1-P(A)$. The complement rule states that the probability of an event not occurring equals one minus the probability of it occurring.
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What is the definition of probability for an event $E$ in a finite equally likely sample space?
What is the definition of probability for an event $E$ in a finite equally likely sample space?
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$P(E)=\frac{#(E)}{#(S)}$. In a finite sample space where all outcomes are equally likely, the probability of event $E$ is the number of favorable outcomes divided by the total number of outcomes.
$P(E)=\frac{#(E)}{#(S)}$. In a finite sample space where all outcomes are equally likely, the probability of event $E$ is the number of favorable outcomes divided by the total number of outcomes.
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What is the probability of at least one head in $3$ fair coin flips?
What is the probability of at least one head in $3$ fair coin flips?
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$\frac{7}{8}$. Calculate as $1 - P($all tails$) = 1 - (\frac{1}{2})^3 = 1 - \frac{1}{8} = \frac{7}{8}$.
$\frac{7}{8}$. Calculate as $1 - P($all tails$) = 1 - (\frac{1}{2})^3 = 1 - \frac{1}{8} = \frac{7}{8}$.
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Identify $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.2$ and $P(B)=0.6$.
Identify $P(A\cap B)$ if $A$ and $B$ are independent with $P(A)=0.2$ and $P(B)=0.6$.
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$0.12$. For independent events, $P(A\cap B) = 0.2 \times 0.6 = 0.12$.
$0.12$. For independent events, $P(A\cap B) = 0.2 \times 0.6 = 0.12$.
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Identify $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.3$.
Identify $P(A\mid B)$ given $P(A\cap B)=0.12$ and $P(B)=0.3$.
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$0.4$. Use the conditional probability formula: $P(A\mid B) = \frac{0.12}{0.3} = 0.4$.
$0.4$. Use the conditional probability formula: $P(A\mid B) = \frac{0.12}{0.3} = 0.4$.
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Identify $P(A\cap B)$ if $P(A)=0.3$ and $P(B\mid A)=0.5$.
Identify $P(A\cap B)$ if $P(A)=0.3$ and $P(B\mid A)=0.5$.
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$0.15$. Apply the multiplication rule: $P(A\cap B) = 0.3 \times 0.5 = 0.15$.
$0.15$. Apply the multiplication rule: $P(A\cap B) = 0.3 \times 0.5 = 0.15$.
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Find $P(A^c)$ given $P(A)=0.13$.
Find $P(A^c)$ given $P(A)=0.13$.
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$0.87$. Use the complement rule: $P(A^c) = 1 - 0.13 = 0.87$.
$0.87$. Use the complement rule: $P(A^c) = 1 - 0.13 = 0.87$.
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Identify $P(A\cup B)$ given $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
Identify $P(A\cup B)$ given $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.
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$0.7$. Apply the addition rule: $P(A\cup B) = 0.4 + 0.5 - 0.2 = 0.7$.
$0.7$. Apply the addition rule: $P(A\cup B) = 0.4 + 0.5 - 0.2 = 0.7$.
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What is the formula for probability of “at least one” occurrence of event $A$ in repeated trials?
What is the formula for probability of “at least one” occurrence of event $A$ in repeated trials?
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$P(\ge 1)=1-P(0)$. The probability of at least one occurrence is one minus the probability of zero occurrences in independent trials.
$P(\ge 1)=1-P(0)$. The probability of at least one occurrence is one minus the probability of zero occurrences in independent trials.
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What is the counting formula for combinations of $n$ distinct items chosen $r$ at a time?
What is the counting formula for combinations of $n$ distinct items chosen $r$ at a time?
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$\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Combinations count the number of ways to choose $r$ items out of $n$ distinct ones, where order does not matter.
$\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Combinations count the number of ways to choose $r$ items out of $n$ distinct ones, where order does not matter.
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Identify $E[X]$ for $X\sim\text{Bin}(10,0.3)$.
Identify $E[X]$ for $X\sim\text{Bin}(10,0.3)$.
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$3$. For binomial, $E[X] = np = 10 \times 0.3 = 3$.
$3$. For binomial, $E[X] = np = 10 \times 0.3 = 3$.
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What is the formula for the variance of $X\sim\text{Bin}(n,p)$?
What is the formula for the variance of $X\sim\text{Bin}(n,p)$?
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$\text{Var}(X)=np(1-p)$. The variance of a binomial random variable measures spread as $np$ times the failure probability.
$\text{Var}(X)=np(1-p)$. The variance of a binomial random variable measures spread as $np$ times the failure probability.
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What is the formula for the expected value of a binomial random variable $X\sim\text{Bin}(n,p)$?
What is the formula for the expected value of a binomial random variable $X\sim\text{Bin}(n,p)$?
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$E[X]=np$. The expected value of a binomial random variable is the product of the number of trials and the success probability per trial.
$E[X]=np$. The expected value of a binomial random variable is the product of the number of trials and the success probability per trial.
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What is the formula for the probability of exactly $k$ successes in $n$ Bernoulli trials?
What is the formula for the probability of exactly $k$ successes in $n$ Bernoulli trials?
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$\binom{n}{k}p^k(1-p)^{n-k}$. The binomial probability formula gives the likelihood of exactly $k$ successes in $n$ independent trials each with success probability $p$.
$\binom{n}{k}p^k(1-p)^{n-k}$. The binomial probability formula gives the likelihood of exactly $k$ successes in $n$ independent trials each with success probability $p$.
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What is $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)$ is known?
What is $P(A\mid B)$ if $A$ and $B$ are independent and $P(A)$ is known?
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$P(A\mid B)=P(A)$. For independent events, the occurrence of $B$ does not affect the probability of $A$, so the conditional equals the unconditional probability.
$P(A\mid B)=P(A)$. For independent events, the occurrence of $B$ does not affect the probability of $A$, so the conditional equals the unconditional probability.
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What condition defines independence of events $A$ and $B$ using an equation?
What condition defines independence of events $A$ and $B$ using an equation?
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$P(A\cap B)=P(A)P(B)$. Events $A$ and $B$ are independent if their joint probability equals the product of their marginal probabilities.
$P(A\cap B)=P(A)P(B)$. Events $A$ and $B$ are independent if their joint probability equals the product of their marginal probabilities.
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What is the formula for conditional probability $P(A\mid B)$?
What is the formula for conditional probability $P(A\mid B)$?
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$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability measures the likelihood of $A$ occurring given $B$ has occurred, by dividing the joint probability by $P(B)$.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Conditional probability measures the likelihood of $A$ occurring given $B$ has occurred, by dividing the joint probability by $P(B)$.
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What is the multiplication rule using conditional probability for events $A$ and $B$?
What is the multiplication rule using conditional probability for events $A$ and $B$?
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$P(A\cap B)=P(A)P(B\mid A)$. The multiplication rule expresses the joint probability as the product of one event's probability and the conditional probability of the other given the first.
$P(A\cap B)=P(A)P(B\mid A)$. The multiplication rule expresses the joint probability as the product of one event's probability and the conditional probability of the other given the first.
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What is the counting formula for permutations of $n$ distinct items taken $r$ at a time?
What is the counting formula for permutations of $n$ distinct items taken $r$ at a time?
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$P(n,r)=\frac{n!}{(n-r)!}$. Permutations count the number of ways to arrange $r$ items out of $n$ distinct ones, where order matters.
$P(n,r)=\frac{n!}{(n-r)!}$. Permutations count the number of ways to arrange $r$ items out of $n$ distinct ones, where order matters.
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What is the probability of getting at least one $6$ in $2$ fair die rolls?
What is the probability of getting at least one $6$ in $2$ fair die rolls?
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$\frac{11}{36}$. Calculate as $1 - P($no 6$) = 1 - (\frac{5}{6})^2 = 1 - \frac{25}{36} = \frac{11}{36}$.
$\frac{11}{36}$. Calculate as $1 - P($no 6$) = 1 - (\frac{5}{6})^2 = 1 - \frac{25}{36} = \frac{11}{36}$.
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What is the probability of drawing $2$ red cards in $2$ draws with replacement from a $52$-card deck?
What is the probability of drawing $2$ red cards in $2$ draws with replacement from a $52$-card deck?
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$\frac{1}{4}$. With replacement, $P($both red$) = \frac{26}{52} \times \frac{26}{52} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
$\frac{1}{4}$. With replacement, $P($both red$) = \frac{26}{52} \times \frac{26}{52} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.
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What is the probability of exactly $2$ heads in $3$ fair coin flips?
What is the probability of exactly $2$ heads in $3$ fair coin flips?
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$\frac{3}{8}$. Use binomial formula: $\binom{3}{2} (\frac{1}{2})^3 = 3 \times \frac{1}{8} = \frac{3}{8}$.
$\frac{3}{8}$. Use binomial formula: $\binom{3}{2} (\frac{1}{2})^3 = 3 \times \frac{1}{8} = \frac{3}{8}$.
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