Probability - PSAT Math
Card 1 of 30
A bag has $3$ red and $2$ blue marbles. Two are drawn without replacement. What is $P(\text{both red})$?
A bag has $3$ red and $2$ blue marbles. Two are drawn without replacement. What is $P(\text{both red})$?
Tap to reveal answer
$\frac{3}{10}$. First red $\frac{3}{5}$, then second red $\frac{2}{4}$: $\frac{3}{5}\times\frac{2}{4}=\frac{3}{10}$.
$\frac{3}{10}$. First red $\frac{3}{5}$, then second red $\frac{2}{4}$: $\frac{3}{5}\times\frac{2}{4}=\frac{3}{10}$.
← Didn't Know|Knew It →
A fair coin is flipped $3$ times. What is $P(\text{at least one head})$?
A fair coin is flipped $3$ times. What is $P(\text{at least one head})$?
Tap to reveal answer
$\frac{7}{8}$. Use complement: $1-P(\text{all tails})=1-\frac{1}{8}=\frac{7}{8}$.
$\frac{7}{8}$. Use complement: $1-P(\text{all tails})=1-\frac{1}{8}=\frac{7}{8}$.
← Didn't Know|Knew It →
What is the complement rule for an event $A$ in probability notation?
What is the complement rule for an event $A$ in probability notation?
Tap to reveal answer
$P(A^c)=1-P(A)$. The probability of not-A equals 1 minus the probability of A.
$P(A^c)=1-P(A)$. The probability of not-A equals 1 minus the probability of A.
← Didn't Know|Knew It →
Find $P(A\cup B)$ if $A$ and $B$ are disjoint, $P(A)=0.25$, and $P(B)=0.40$.
Find $P(A\cup B)$ if $A$ and $B$ are disjoint, $P(A)=0.25$, and $P(B)=0.40$.
Tap to reveal answer
$0.65$. Disjoint events: simply add $0.25+0.40=0.65$.
$0.65$. Disjoint events: simply add $0.25+0.40=0.65$.
← Didn't Know|Knew It →
Find $P(A\cap B)$ if $P(A)=0.60$ and $P(B\mid A)=0.20$.
Find $P(A\cap B)$ if $P(A)=0.60$ and $P(B\mid A)=0.20$.
Tap to reveal answer
$0.12$. Apply multiplication rule: $0.60\times 0.20=0.12$.
$0.12$. Apply multiplication rule: $0.60\times 0.20=0.12$.
← Didn't Know|Knew It →
What is the addition rule for any events $A$ and $B$ (not necessarily disjoint)?
What is the addition rule for any events $A$ and $B$ (not necessarily disjoint)?
Tap to reveal answer
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Subtract overlap to avoid double-counting when adding probabilities.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Subtract overlap to avoid double-counting when adding probabilities.
← Didn't Know|Knew It →
What is the probability of exactly $k$ successes in $n$ independent trials with success rate $p$?
What is the probability of exactly $k$ successes in $n$ independent trials with success rate $p$?
Tap to reveal answer
$\binom{n}{k}p^k(1-p)^{n-k}$. Binomial probability formula for k successes in n trials.
$\binom{n}{k}p^k(1-p)^{n-k}$. Binomial probability formula for k successes in n trials.
← Didn't Know|Knew It →
Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.30$.
Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.30$.
Tap to reveal answer
$0.60$. Apply conditional formula: $\frac{0.18}{0.30}=0.60$.
$0.60$. Apply conditional formula: $\frac{0.18}{0.30}=0.60$.
← Didn't Know|Knew It →
What is the addition rule for disjoint events $A$ and $B$?
What is the addition rule for disjoint events $A$ and $B$?
Tap to reveal answer
If $A\cap B=\varnothing$, then $P(A\cup B)=P(A)+P(B)$. Disjoint events have no overlap, so just add their probabilities.
If $A\cap B=\varnothing$, then $P(A\cup B)=P(A)+P(B)$. Disjoint events have no overlap, so just add their probabilities.
← Didn't Know|Knew It →
What is the multiplication rule for probability using conditional probability?
What is the multiplication rule for probability using conditional probability?
Tap to reveal answer
$P(A\cap B)=P(A),P(B\mid A)$. Multiply the probability of A by the probability of B given A.
$P(A\cap B)=P(A),P(B\mid A)$. Multiply the probability of A by the probability of B given A.
← Didn't Know|Knew It →
A bag has $3$ red and $2$ blue marbles. One is drawn. What is $P(\text{blue})$?
A bag has $3$ red and $2$ blue marbles. One is drawn. What is $P(\text{blue})$?
Tap to reveal answer
$\frac{2}{5}$. 2 blue marbles out of 5 total: $\frac{2}{5}$.
$\frac{2}{5}$. 2 blue marbles out of 5 total: $\frac{2}{5}$.
← Didn't Know|Knew It →
What is the probability rule for “at least one” occurrence of event $A$ in $n$ trials?
What is the probability rule for “at least one” occurrence of event $A$ in $n$ trials?
Tap to reveal answer
$P(\ge 1)=1-P(0)$. At least one equals 1 minus the probability of none.
$P(\ge 1)=1-P(0)$. At least one equals 1 minus the probability of none.
← Didn't Know|Knew It →
What is the probability of an event when outcomes are equally likely?
What is the probability of an event when outcomes are equally likely?
Tap to reveal answer
$P(E)=\frac{\text{favorable}}{\text{total}}$. Count favorable outcomes and divide by total possible outcomes.
$P(E)=\frac{\text{favorable}}{\text{total}}$. Count favorable outcomes and divide by total possible outcomes.
← Didn't Know|Knew It →
Two fair coins are flipped. What is $P(\text{exactly one head})$?
Two fair coins are flipped. What is $P(\text{exactly one head})$?
Tap to reveal answer
$\frac{1}{2}$. Outcomes HT and TH give exactly one head: $\frac{2}{4}=\frac{1}{2}$.
$\frac{1}{2}$. Outcomes HT and TH give exactly one head: $\frac{2}{4}=\frac{1}{2}$.
← Didn't Know|Knew It →
What is the expected value formula for a discrete random variable $X$?
What is the expected value formula for a discrete random variable $X$?
Tap to reveal answer
$E(X)=\sum x,P(X=x)$. Sum each value times its probability for discrete variables.
$E(X)=\sum x,P(X=x)$. Sum each value times its probability for discrete variables.
← Didn't Know|Knew It →
A fair die is rolled once. What is $P(\text{roll is a multiple of }3)$?
A fair die is rolled once. What is $P(\text{roll is a multiple of }3)$?
Tap to reveal answer
$\frac{1}{3}$. Multiples of 3 are {3,6}, so $\frac{2}{6}=\frac{1}{3}$.
$\frac{1}{3}$. Multiples of 3 are {3,6}, so $\frac{2}{6}=\frac{1}{3}$.
← Didn't Know|Knew It →
What is the formula for conditional probability $P(A\mid B)$ (with $P(B)>0$)?
What is the formula for conditional probability $P(A\mid B)$ (with $P(B)>0$)?
Tap to reveal answer
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Divide the joint probability by the condition's probability.
$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Divide the joint probability by the condition's probability.
← Didn't Know|Knew It →
A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then blue})$?
A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then blue})$?
Tap to reveal answer
$\frac{6}{25}$. $\frac{3}{5}\times\frac{2}{5}$ since marble is replaced.
$\frac{6}{25}$. $\frac{3}{5}\times\frac{2}{5}$ since marble is replaced.
← Didn't Know|Knew It →
If $P(A)=0.3$ and $P(B)=0.4$ are independent, what is $P(A\cap B)$?
If $P(A)=0.3$ and $P(B)=0.4$ are independent, what is $P(A\cap B)$?
Tap to reveal answer
$0.12$. Multiply independent probabilities: $0.3\times 0.4$.
$0.12$. Multiply independent probabilities: $0.3\times 0.4$.
← Didn't Know|Knew It →
What is $P(\text{not }5)$ when rolling one fair six-sided die?
What is $P(\text{not }5)$ when rolling one fair six-sided die?
Tap to reveal answer
$\frac{5}{6}$. Five favorable outcomes (1,2,3,4,6) out of six.
$\frac{5}{6}$. Five favorable outcomes (1,2,3,4,6) out of six.
← Didn't Know|Knew It →
What is the formula for the complement of an event $A$?
What is the formula for the complement of an event $A$?
Tap to reveal answer
$P(A^c)=1-P(A)$. The complement has probability one minus the original.
$P(A^c)=1-P(A)$. The complement has probability one minus the original.
← Didn't Know|Knew It →
What is the range of possible values for any probability $P(A)$?
What is the range of possible values for any probability $P(A)$?
Tap to reveal answer
$0\le P(A)\le 1$. Probabilities must be between 0 and 1 inclusive.
$0\le P(A)\le 1$. Probabilities must be between 0 and 1 inclusive.
← Didn't Know|Knew It →
Identify the probability of an impossible event.
Identify the probability of an impossible event.
Tap to reveal answer
$0$. Impossible events have zero probability.
$0$. Impossible events have zero probability.
← Didn't Know|Knew It →
Identify the probability of a certain event.
Identify the probability of a certain event.
Tap to reveal answer
$1$. Certain events always occur with probability one.
$1$. Certain events always occur with probability one.
← Didn't Know|Knew It →
What is $P(\text{even})$ when rolling one fair six-sided die?
What is $P(\text{even})$ when rolling one fair six-sided die?
Tap to reveal answer
$\frac{1}{2}$. Three even outcomes (2,4,6) out of six total.
$\frac{1}{2}$. Three even outcomes (2,4,6) out of six total.
← Didn't Know|Knew It →
What is $P(\text{at least one head})$ when flipping $2$ fair coins?
What is $P(\text{at least one head})$ when flipping $2$ fair coins?
Tap to reveal answer
$\frac{3}{4}$. Three favorable (HH, HT, TH) out of four outcomes.
$\frac{3}{4}$. Three favorable (HH, HT, TH) out of four outcomes.
← Didn't Know|Knew It →
A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ in one draw?
A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ in one draw?
Tap to reveal answer
$\frac{3}{5}$. Three red marbles out of five total marbles.
$\frac{3}{5}$. Three red marbles out of five total marbles.
← Didn't Know|Knew It →
A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then blue})$?
A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then blue})$?
Tap to reveal answer
$\frac{3}{10}$. $\frac{3}{5}\times\frac{2}{4}$ since one red is removed.
$\frac{3}{10}$. $\frac{3}{5}\times\frac{2}{4}$ since one red is removed.
← Didn't Know|Knew It →
What is $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.3$?
What is $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.3$?
Tap to reveal answer
$0.4$. Conditional probability: $\frac{0.12}{0.3} = 0.4$.
$0.4$. Conditional probability: $\frac{0.12}{0.3} = 0.4$.
← Didn't Know|Knew It →
What is the addition rule for any two events $A$ and $B$?
What is the addition rule for any two events $A$ and $B$?
Tap to reveal answer
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Add individual probabilities, subtract overlap to avoid double-counting.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Add individual probabilities, subtract overlap to avoid double-counting.
← Didn't Know|Knew It →