Basic Probability - ISEE Middle Level: Quantitative Reasoning
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A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ on one draw?
A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ on one draw?
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$\frac{3}{5}$. There are 3 red marbles out of 5 total marbles in the bag.
$\frac{3}{5}$. There are 3 red marbles out of 5 total marbles in the bag.
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A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then red})$?
A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then red})$?
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$\frac{3}{10}$. Without replacement, multiply $\frac{3}{5}$ by $\frac{2}{4}$ for two red draws.
$\frac{3}{10}$. Without replacement, multiply $\frac{3}{5}$ by $\frac{2}{4}$ for two red draws.
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A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then red})$?
A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then red})$?
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$\frac{9}{25}$. With replacement, the probability remains $\frac{3}{5}$ for each independent draw, so multiply.
$\frac{9}{25}$. With replacement, the probability remains $\frac{3}{5}$ for each independent draw, so multiply.
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What is the probability of getting heads on a fair coin flip?
What is the probability of getting heads on a fair coin flip?
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$\frac{1}{2}$. One favorable outcome (heads) out of two equally likely outcomes on a fair coin.
$\frac{1}{2}$. One favorable outcome (heads) out of two equally likely outcomes on a fair coin.
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What is the probability of an event $E$ if it has $f$ favorable outcomes out of $n$ equally likely outcomes?
What is the probability of an event $E$ if it has $f$ favorable outcomes out of $n$ equally likely outcomes?
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$P(E)=\frac{f}{n}$. Probability is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes.
$P(E)=\frac{f}{n}$. Probability is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes.
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What is the probability of an event that is impossible (has $0$ favorable outcomes)?
What is the probability of an event that is impossible (has $0$ favorable outcomes)?
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$0$. An impossible event has no favorable outcomes, resulting in a probability of zero.
$0$. An impossible event has no favorable outcomes, resulting in a probability of zero.
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What is the probability of an event that is certain (always occurs)?
What is the probability of an event that is certain (always occurs)?
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$1$. A certain event includes all possible outcomes, so its probability equals one.
$1$. A certain event includes all possible outcomes, so its probability equals one.
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What is the relationship between an event $E$ and its complement $E^c$?
What is the relationship between an event $E$ and its complement $E^c$?
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$P(E^c)=1-P(E)$. The probability of the complement is one minus the event's probability, as they together cover all outcomes.
$P(E^c)=1-P(E)$. The probability of the complement is one minus the event's probability, as they together cover all outcomes.
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What is $P(E)$ if $P(E^c)=\frac{3}{8}$?
What is $P(E)$ if $P(E^c)=\frac{3}{8}$?
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$\frac{5}{8}$. Subtract the probability of the complement from 1 to find the event's probability.
$\frac{5}{8}$. Subtract the probability of the complement from 1 to find the event's probability.
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What is $P(E^c)$ if $P(E)=0.27$?
What is $P(E^c)$ if $P(E)=0.27$?
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$0.73$. The complement's probability is 1 minus the event's probability.
$0.73$. The complement's probability is 1 minus the event's probability.
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What is the probability of rolling a $4$ on a fair six-sided die?
What is the probability of rolling a $4$ on a fair six-sided die?
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$\frac{1}{6}$. One favorable outcome (rolling a 4) out of six equally likely outcomes on a fair die.
$\frac{1}{6}$. One favorable outcome (rolling a 4) out of six equally likely outcomes on a fair die.
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What is the probability of rolling an even number on a fair six-sided die?
What is the probability of rolling an even number on a fair six-sided die?
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$\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes on a fair die.
$\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes on a fair die.
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What is the probability of getting at least one head in two fair coin flips?
What is the probability of getting at least one head in two fair coin flips?
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$\frac{3}{4}$. Calculated as 1 minus the probability of no heads (both tails), which is $\frac{1}{4}$.
$\frac{3}{4}$. Calculated as 1 minus the probability of no heads (both tails), which is $\frac{1}{4}$.
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What is the probability of getting two heads in two fair coin flips?
What is the probability of getting two heads in two fair coin flips?
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$\frac{1}{4}$. The probability of heads on each independent flip is $\frac{1}{2}$, so multiply for both.
$\frac{1}{4}$. The probability of heads on each independent flip is $\frac{1}{2}$, so multiply for both.
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What is the probability of drawing a heart from a standard $52$-card deck?
What is the probability of drawing a heart from a standard $52$-card deck?
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$\frac{1}{4}$. There are 13 hearts out of 52 cards in a standard deck.
$\frac{1}{4}$. There are 13 hearts out of 52 cards in a standard deck.
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What is the probability of drawing an ace from a standard $52$-card deck?
What is the probability of drawing an ace from a standard $52$-card deck?
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$\frac{1}{13}$. There are 4 aces out of 52 cards in a standard deck.
$\frac{1}{13}$. There are 4 aces out of 52 cards in a standard deck.
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What is the probability of drawing a red card from a standard $52$-card deck?
What is the probability of drawing a red card from a standard $52$-card deck?
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$\frac{1}{2}$. There are 26 red cards (hearts and diamonds) out of 52 in a standard deck.
$\frac{1}{2}$. There are 26 red cards (hearts and diamonds) out of 52 in a standard deck.
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What is the probability of drawing a face card (J, Q, or K) from a $52$-card deck?
What is the probability of drawing a face card (J, Q, or K) from a $52$-card deck?
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$\frac{3}{13}$. There are 12 face cards (3 per suit, 4 suits) out of 52 in a standard deck.
$\frac{3}{13}$. There are 12 face cards (3 per suit, 4 suits) out of 52 in a standard deck.
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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, the union's probability is the sum since there is no overlap.
$P(A\cup B)=P(A)+P(B)$. For mutually exclusive events, the union's probability is the sum since there is no overlap.
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If events $A$ and $B$ are mutually exclusive, what is $P(A\cap B)$?
If events $A$ and $B$ are mutually exclusive, what is $P(A\cap B)$?
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$0$. Mutually exclusive events cannot occur together, so their intersection has zero probability.
$0$. Mutually exclusive events cannot occur together, so their intersection has zero probability.
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If $P(A)=\frac{1}{5}$ and $P(B)=\frac{2}{5}$ and $A,B$ are mutually exclusive, what is $P(A\cup B)$?
If $P(A)=\frac{1}{5}$ and $P(B)=\frac{2}{5}$ and $A,B$ are mutually exclusive, what is $P(A\cup B)$?
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$\frac{3}{5}$. Add the probabilities since the events are mutually exclusive with no overlap.
$\frac{3}{5}$. Add the probabilities since the events are mutually exclusive with no overlap.
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What is the general addition rule for any events $A$ and $B$ (not necessarily disjoint)?
What is the general addition rule for any events $A$ and $B$ (not necessarily disjoint)?
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$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The general rule accounts for overlap by subtracting the intersection's probability.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. The general rule accounts for overlap by subtracting the intersection's probability.
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If $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?
If $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?
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$0.9$. Apply the general addition rule: add probabilities and subtract the intersection.
$0.9$. Apply the general addition rule: add probabilities and subtract the intersection.
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What is the multiplication rule for independent events $A$ and $B$?
What is the multiplication rule for independent events $A$ and $B$?
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$P(A\cap B)=P(A)P(B)$. For independent events, the joint probability is the product of individual probabilities.
$P(A\cap B)=P(A)P(B)$. For independent events, the joint probability is the product of individual probabilities.
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If $P(A)=\frac{3}{10}$ and $P(B)=\frac{1}{2}$ and $A,B$ are independent, what is $P(A\cap B)$?
If $P(A)=\frac{3}{10}$ and $P(B)=\frac{1}{2}$ and $A,B$ are independent, what is $P(A\cap B)$?
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$\frac{3}{20}$. Multiply the probabilities since the events are independent.
$\frac{3}{20}$. Multiply the probabilities since the events are independent.
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