PSAT Math Flashcards: Probability

What this deck covers

This deck focuses on Probability, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What condition must hold for events $A$ and $B$ to be independent using conditional probability?

Answer: $P(A\mid B)=P(A)$. Independence means $B$ doesn't change $A$'s probability.

Flashcard 2: What is the formula for the expected value of a discrete variable with outcomes $x_i$ and probabilities $p_i$?

Answer: $E=\sum x_i p_i$. Sum of each outcome times its probability.

Flashcard 3: A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ in one draw?

Answer: $\frac{3}{5}$. Three red marbles out of five total marbles.

Flashcard 4: What is the complement rule for an event $A$?

Answer: $P(A^c)=1-P(A)$. The probability of not-$A$ equals one minus the probability of $A$.

Flashcard 5: What is the definition formula for conditional probability $P(A\mid B)$?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Ratio of joint probability to the condition's probability.

Flashcard 6: Find $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.3$, and $P(B)=0.4$.

Answer: $0.12$. For independent events: $P(A \cap B) = 0.3 \times 0.4 = 0.12$.

Flashcard 7: What is the probability of a union if $P(A)=0.3$, $P(B)=0.5$, and $P(A\cap B)=0.1$?

Answer: $P(A\cup B)=0.7$. Apply general addition rule: $0.3+0.5-0.1=0.7$.

Flashcard 8: What is $P(A\cup B)$ if $P(A)=0.5$, $P(B)=0.6$, and $P(A\cap B)=0.2$?

Answer: $0.9$. Apply general addition: $0.5 + 0.6 - 0.2 = 0.9$.

Flashcard 9: Find $P(A\cup B)$ if $A$ and $B$ are disjoint, $P(A)=0.25$, and $P(B)=0.40$.

Answer: $0.65$. Disjoint events: simply add $0.25+0.40=0.65$.

Flashcard 10: What is the formula connecting conditional probability to intersection: $P(A\cap B)$?

Answer: $P(A\cap B)=P(A\mid B)\cdot P(B)$. Rearranges conditional probability to find intersection.

Flashcard 11: What is the formula for probability of event $A$ using favorable and total outcomes?

Answer: $P(A)=\frac{\text{favorable}}{\text{total}}$. Divide favorable outcomes by total possible outcomes.

Flashcard 12: What is the expected value of a game paying $5 with probability $0.2$ and $0 otherwise?

Answer: $1$. Expected value: $5\times 0.2+0\times 0.8=1$.

Flashcard 13: What is the probability of flipping exactly one head in two fair coin flips?

Answer: $\frac{1}{2}$. Outcomes HT and TH out of HH, HT, TH, TT.

Flashcard 14: What is the probability range for any event $E$?

Answer: $0\le P(E)\le 1$. Probability is always between impossible (0) and certain (1).

Flashcard 15: Identify whether $A$ and $B$ are independent if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

Answer: Independent. Check: $P(A) \times P(B) = 0.4 \times 0.5 = 0.2 = P(A \cap B)$.

Flashcard 16: What is the addition rule for disjoint events $A$ and $B$?

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.

Flashcard 17: A bag has $5$ red and $3$ blue marbles. One marble is drawn. What is $P(\text{red})$?

Answer: $\frac{5}{8}$. Five red marbles out of eight total marbles.

Flashcard 18: What is the range of possible values for any probability $P(A)$?

Answer: $0\le P(A)\le 1$. Probabilities must be between 0 and 1 inclusive.

Flashcard 19: What is $P(\text{not }5)$ when rolling one fair six-sided die?

Answer: $\frac{5}{6}$. Five favorable outcomes (1,2,3,4,6) out of six.

Flashcard 20: A bag has $5$ red and $3$ blue marbles. What is $P(\text{red})$ on one draw?

Answer: $\frac{5}{8}$. 5 red marbles out of 8 total marbles.

Flashcard 21: A bag has $5$ red and $3$ blue marbles. Two are drawn with replacement. What is $P(\text{both red})$?

Answer: $\left(\frac{5}{8}\right)^2=\frac{25}{64}$. With replacement, each draw has $\frac{5}{8}$ probability; multiply for both.

Flashcard 22: What is the formula for the complement rule for an event $A$?

Answer: $P(A^c)=1-P(A)$. The probability of not-$A$ equals one minus the probability of $A$.

Flashcard 23: A fair die is rolled once. What is $P(\text{roll is a multiple of }3)$?

Answer: $\frac{1}{3}$. Multiples of 3 are {3,6}, so $\frac{2}{6}=\frac{1}{3}$.

Flashcard 24: Find $P(A\cup B)$ if $P(A)=0.5$, $P(B)=0.3$, and $P(A\cap B)=0.1$.

Answer: $0.7$. Use addition rule: $0.5+0.3-0.1=0.7$.

Flashcard 25: Find $P(A\cup B)$ if $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

Answer: $0.9$. Apply general addition rule: $0.6 + 0.5 - 0.2 = 0.9$.

Flashcard 26: What is $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.3$, and $P(B)=0.6$?

Answer: $0.18$. Independent events: $0.3\times 0.6=0.18$.

Flashcard 27: A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then blue})$?

Answer: $\frac{6}{25}$. $\frac{3}{5}\times\frac{2}{5}$ since marble is replaced.

Flashcard 28: A bag has $3$ red and $2$ blue marbles. Two are drawn without replacement. What is $P(\text{both red})$?

Answer: $\frac{3}{10}$. First red $\frac{3}{5}$, then second red $\frac{2}{4}$: $\frac{3}{5}\times\frac{2}{4}=\frac{3}{10}$.

Flashcard 29: What does it mean for events $A$ and $B$ to be mutually exclusive?

Answer: $P(A\cap B)=0$. Mutually exclusive events cannot occur simultaneously.

Flashcard 30: Find $P(A\mid B)$ if $P(A\cap B)=0.15$ and $P(B)=0.5$.

Answer: $0.3$. Apply conditional probability: $P(A|B) = \frac{0.15}{0.5} = 0.3$.

Flashcard 31: If $P(A)=0.3$, $P(B)=0.5$, and $A$ and $B$ are independent, what is $P(A\cap B)$?

Answer: $0.15$. For independent events, $P(A\cap B)=P(A)\cdot P(B)=0.3\times 0.5$.

Flashcard 32: What condition defines independence using probabilities of $A$ and $B$?

Answer: $A,B$ independent if $P(A\cap B)=P(A)P(B)$. Events are independent when their joint probability equals the product.

Flashcard 33: What is the formula for conditional probability $P(A\mid B)$ (with $P(B)>0$)?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Divide the joint probability by the condition's probability.

Flashcard 34: What is $P(A\cap B)$ if $P(A)=0.3$ and $P(B)=0.4$ and the events are independent?

Answer: $0.12$. Independent events: $0.3\times 0.4=0.12$.

Flashcard 35: What is $P(\text{at least one head})$ when flipping $2$ fair coins?

Answer: $\frac{3}{4}$. Three favorable (HH, HT, TH) out of four outcomes.

Flashcard 36: What is the conditional probability formula $P(A\mid B)$?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Divides joint probability by the condition's probability.

Flashcard 37: Identify the event described by "$A$ and $B$" using set notation.

Answer: $A\cap B$. Intersection symbol represents "and" in probability.

Flashcard 38: What is $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.3$, and $P(B)=0.4$?

Answer: $0.12$. Independent events: $P(A\cap B) = 0.3 \times 0.4 = 0.12$.

Flashcard 39: A fair die is rolled. What is the probability of rolling a number greater than $4$?

Answer: $\frac{1}{3}$. Favorable outcomes are 5 and 6, so $\frac{2}{6}=\frac{1}{3}$.

Flashcard 40: If $P(A)=0.35$, what is $P(A^c)$?

Answer: $0.65$. Using complement rule: $1-0.35=0.65$

Flashcard 41: A bag has $3$ red and $2$ blue marbles. Two are drawn without replacement. What is $P(\text{both red})$?

Answer: $\frac{3}{10}$. First red $\frac{3}{5}$, then $\frac{2}{4}$ remaining red: $\frac{3}{5}\times\frac{2}{4}=\frac{3}{10}$.

Flashcard 42: What is the multiplication rule for any events $A$ and $B$ using conditional probability?

Answer: $P(A\cap B)=P(B)P(A\mid B)$. General multiplication rule using conditional probability.

Flashcard 43: What is the probability of drawing two aces without replacement from a $52$-card deck?

Answer: $\frac{1}{221}$. $\frac{4}{52}\times\frac{3}{51}=\frac{12}{2652}=\frac{1}{221}$.

Flashcard 44: What is the conditional probability formula for $P(A\mid B)$?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Probability of A given B equals their intersection over B.

Flashcard 45: Find $P(A\cap B)$ if $P(A)=0.3$ and $P(B\mid A)=0.5$.

Answer: $0.15$. Use multiplication rule: $0.3\times 0.5=0.15$.

Flashcard 46: What is the fundamental counting principle for $m$ choices then $n$ choices?

Answer: $m\cdot n$ total outcomes. Multiply the number of choices at each step.

Flashcard 47: What is $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.30$?

Answer: $0.40$. Divide joint probability by condition: $\frac{0.12}{0.30}=0.40$.

Flashcard 48: Which option equals $^5C_2$ (combinations of $5$ items chosen $2$)?

Answer: $10$. $^5C_2 = \frac{5!}{2!3!} = \frac{20}{2} = 10$ ways to choose 2 from 5.

Flashcard 49: What is the probability that a fair die roll is even?

Answer: $\frac{1}{2}$. Three even outcomes (2,4,6) out of six total outcomes.

Flashcard 50: What is the multiplication rule using conditional probability for events $A$ and $B$?

Answer: $P(A\cap B)=P(A\mid B)\cdot P(B)$. Rearranges conditional probability definition to find intersection probability.

Flashcard 51: What is the probability of exactly $k$ successes in $n$ independent trials with success rate $p$?

Answer: $\binom{n}{k}p^k(1-p)^{n-k}$. Binomial probability formula for k successes in n trials.

Flashcard 52: A bag has $3$ red and $2$ blue marbles. Without replacement, what is $P(\text{red then blue})$?

Answer: $\frac{3}{10}$. $P(\text{red})\times P(\text{blue}|\text{red}) = \frac{3}{5}\times\frac{2}{4} = \frac{3}{10}$.

Flashcard 53: What is the probability that a fair die roll is greater than $4$?

Answer: $\frac{1}{3}$. Two favorable outcomes (5,6) out of six total outcomes.

Flashcard 54: What is the probability rule for "at least one" occurrence of event $A$ in $n$ trials?

Answer: $P(\ge 1)=1-P(0)$. At least one equals 1 minus the probability of none.

Flashcard 55: Identify whether $A$ and $B$ are mutually exclusive if $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$.

Answer: Not mutually exclusive. Since $P(A\cap B)=0.2\neq 0$, events can occur together.

Flashcard 56: Find the probability of at least one head in $2$ fair coin flips.

Answer: $\frac{3}{4}$. All outcomes except TT: $\frac{3}{4}$ (HH, HT, TH).

Flashcard 57: What is the formula for permutations of $n$ distinct items taken $r$ at a time?

Answer: $\frac{n!}{(n-r)!}$. Order matters; divide $n!$ by $(n-r)!$ to remove unused items.

Flashcard 58: A fair coin is flipped twice. What is $P(\text{exactly one head})$?

Answer: $\frac{1}{2}$. Outcomes HT and TH out of four equally likely: $\frac{2}{4} = \frac{1}{2}$.

Flashcard 59: Find $P(A^c)$ if $P(A)=0.37$.

Answer: $0.63$. Apply complement rule: $P(A^c) = 1 - 0.37 = 0.63$.

Flashcard 60: A fair die is rolled once. What is $P(\text{prime})$?

Answer: $\frac{1}{2}$. Prime numbers on a die are $2$, $3$, and $5$: three out of six outcomes.

Flashcard 61: What is the formula for probability of an event $A$ using favorable and total outcomes?

Answer: $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Ratio of successful outcomes to all possible outcomes.

Flashcard 62: What is the formula for the complement of an event $A$?

Answer: $P(A^c)=1-P(A)$. The complement has probability one minus the original.

Flashcard 63: If $P(A\cap B)=0.12$ and $P(B)=0.3$, what is $P(A\mid B)$?

Answer: $0.4$. Apply conditional probability formula: $\frac{0.12}{0.3} = 0.4$.

Flashcard 64: A bag has $3$ red and $2$ blue marbles. With replacement, what is $P(\text{red then blue})$?

Answer: $\frac{6}{25}$. With replacement: $P(\text{red})\cdot P(\text{blue})=\frac{3}{5}\cdot\frac{2}{5}=\frac{6}{25}$

Flashcard 65: Find $P(A\cap B)$ if $P(A)=0.60$ and $P(B\mid A)=0.20$.

Answer: $0.12$. Apply multiplication rule: $0.60\times 0.20=0.12$.

Flashcard 66: Find $P(A\cup B)$ if $P(A)=0.5$, $P(B)=0.4$, and $P(A\cap B)=0.2$.

Answer: $0.7$. Apply union formula: $0.5+0.4-0.2=0.7$.

Flashcard 67: What is the counting formula for permutations of $n$ items taken $r$ at a time?

Answer: $^nP_r=\frac{n!}{(n-r)!}$. Order matters: arrange r items from n total items.

Flashcard 68: What is the multiplication rule using conditional probability for events $A$ and $B$?

Answer: $P(A\cap B)=P(B)\cdot P(A\mid B)$. Rearranges conditional probability formula to find intersection.

Flashcard 69: What condition must hold for events $A$ and $B$ to be mutually exclusive (disjoint)?

Answer: $P(A\cap B)=0$. Disjoint events cannot occur simultaneously.

Flashcard 70: How many ways can you choose $2$ students from $5$ students?

Answer: $\binom{5}{2}=10$. Combinations formula: $\frac{5!}{2!3!}=10$.

Flashcard 71: What is the probability of the empty event $\varnothing$?

Answer: $P(\varnothing)=0$. The impossible event has probability 0.

Flashcard 72: If $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

Answer: $0.9$. Apply addition rule: $0.6+0.5-0.2=0.9$.

Flashcard 73: What is the definition of probability for an event with equally likely outcomes?

Answer: $P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Ratio of successful outcomes to all possible outcomes.

Flashcard 74: What is the complement rule for an event $E$?

Answer: $P(E^c)=1-P(E)$. The complement's probability is one minus the event's probability.

Flashcard 75: What is the counting formula for combinations of $n$ distinct items chosen $r$ at a time?

Answer: $\frac{n!}{r!(n-r)!}$. Order doesn't matter; divide by $r!$ to remove arrangements.

Flashcard 76: A bag has $3$ red and $2$ blue marbles. What is $P(\text{red})$ on one draw?

Answer: $\frac{3}{5}$. Three red marbles out of five total marbles.

Flashcard 77: If $P(A)=0.6$, $P(B)=0.5$, and $A$ and $B$ are independent, what is $P(A\cap B)$?

Answer: $0.3$. For independent events: $0.6\times 0.5=0.3$

Flashcard 78: What is the key difference: independent vs. mutually exclusive events (in one statement)?

Answer: Independent: $P(A\cap B)=P(A)P(B)$; disjoint: $P(A\cap B)=0$. Independent can overlap; disjoint cannot occur together.

Flashcard 79: What is the definition of probability of event $A$ using outcomes and favorable outcomes?

Answer: $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Ratio of successful outcomes to all possible outcomes.

Flashcard 80: What is the probability of rolling an even number on a fair six-sided die?

Answer: $\frac{1}{2}$. Three even outcomes (2,4,6) out of six total outcomes.

Flashcard 81: A bag has $3$ red and $2$ blue marbles. Two are drawn with replacement. What is $P(\text{both red})$?

Answer: $\frac{9}{25}$. With replacement: $\frac{3}{5}\times\frac{3}{5}=\frac{9}{25}$.

Flashcard 82: What is the formula for the probability of $A\cup B$ for any two events $A$ and $B$?

Answer: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Add individual probabilities, subtract overlap to avoid double-counting.

Flashcard 83: What condition must hold for events $A$ and $B$ to be mutually exclusive (disjoint)?

Answer: $P(A\cap B)=0$. Mutually exclusive events cannot occur together.

Flashcard 84: A fair die is rolled once. What is $P(\text{prime})$ where primes are $2,3,5$?

Answer: $\frac{1}{2}$. Three primes $(2,3,5)$ out of six outcomes: $\frac{3}{6}=\frac{1}{2}$.

Flashcard 85: What is the probability of not rolling a $6$ on one fair die roll?

Answer: $\frac{5}{6}$. Five favorable outcomes out of six total.

Flashcard 86: What condition must hold for events $A$ and $B$ to be independent?

Answer: $P(A\mid B)=P(A)$. Independence means conditioning doesn't change the probability.

Flashcard 87: What is the definition of conditional probability $P(A\mid B)$?

Answer: $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Probability of A given B equals their intersection divided by B's probability.

Flashcard 88: What is $P(A^c)$ if $P(A)=0.37$?

Answer: $0.63$. Apply complement rule: $1-0.37=0.63$.

Flashcard 89: What condition defines mutual exclusivity (disjointness) for events $A$ and $B$?

Answer: Mutually exclusive if $P(A\cap B)=0$. Events cannot occur together when their intersection is empty.

Flashcard 90: A card is drawn from a standard $52$-card deck. What is $P(\text{heart})$?

Answer: $\frac{1}{4}$. There are $13$ hearts in a standard $52$-card deck.

Flashcard 91: What is the complement rule for an event $A$ in probability notation?

Answer: $P(A^c)=1-P(A)$. The probability of not-A equals 1 minus the probability of A.

Flashcard 92: What is the condition for events $A$ and $B$ to be mutually exclusive?

Answer: $P(A\cap B)=0$. Mutually exclusive events cannot happen simultaneously.

Flashcard 93: Identify whether $A$ and $B$ are independent if $P(A)=0.5$, $P(B)=0.2$, and $P(A\cap B)=0.1$.

Answer: Independent. Check if $P(A\cap B)=P(A)P(B)$: $0.1=0.5×0.2$ ✓.

Flashcard 94: If events $A$ and $B$ are mutually exclusive with $P(A)=0.3$ and $P(B)=0.5$, what is $P(A\cup B)$?

Answer: $0.8$. For mutually exclusive events, add probabilities: $0.3 + 0.5 = 0.8$.

Flashcard 95: What is the expected value formula for a discrete random variable $X$?

Answer: $E(X)=\sum x\,P(X=x)$. Sum each value times its probability for discrete variables.

Flashcard 96: Find the probability of exactly one head in $2$ fair coin flips.

Answer: $\frac{1}{2}$. Outcomes HT and TH out of 4 total: $\frac{2}{4} = \frac{1}{2}$.

Flashcard 97: Which condition must hold for events $A$ and $B$ to be independent (in probability form)?

Answer: $P(A\mid B)=P(A)$. Independent events satisfy: knowing $B$ occurred doesn't change $P(A)$.

Flashcard 98: A fair coin is flipped $3$ times. What is the probability of exactly $2$ heads?

Answer: $\frac{3}{8}$. Use binomial: $\binom{3}{2}(\frac{1}{2})^2(\frac{1}{2})^1=\frac{3}{8}$.

Flashcard 99: Two fair dice are rolled. What is $P(\text{sum}=7)$?

Answer: $\frac{1}{6}$. Six ways to sum to $7$: $(1,6)$, $(2,5)$, $(3,4)$, $(4,3)$, $(5,2)$, $(6,1)$ out of $36$.

Flashcard 100: Find $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.3$.

Answer: $0.4$. Divide intersection by condition: $\frac{0.12}{0.3}=0.4$.