ACT 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 ACT 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 is the total probability formula for a sample space S?

Answer: P(S) = 1. All possible outcomes in a sample space sum to probability 1.

Flashcard 2: What is the probability of drawing a king or a queen from a deck?

Answer: $\frac{2}{13}$. 8 cards (4 kings + 4 queens) out of 52 total cards.

Flashcard 3: Calculate the probability of not rolling a 5 on a six-sided die.

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

Flashcard 4: What is the probability of the complement of $A\cup B$ in terms of $P(A\cup B)$?

Answer: $P((A\cup B)^c)=1-P(A\cup B)$. Complement rule applied to the union of events.

Flashcard 5: What is $P(\text{sum }7)$ when rolling two fair six-sided dice?

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

Flashcard 6: What is the complement rule for an event $E$?

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

Flashcard 7: Calculate the probability of rolling a 2 or a 5 on a six-sided die.

Answer: $\frac{1}{3}$. Two favorable outcomes (2 or 5) out of six possible.

Flashcard 8: If $P(A)=0.2$ and $P(B)=0.6$ and $A,B$ are independent, what is $P(A\cup B)$?

Answer: $0.68$. For independent events: $P(A\cup B) = 0.2 + 0.6 - (0.2)(0.6)$.

Flashcard 9: What is the probability of a certain event (an event that must happen)?

Answer: $1$. An event that must occur has probability 1.

Flashcard 10: What is the probability of a certain event (an event that must happen)?

Answer: $1$. An event that must occur has probability 1.

Flashcard 11: From a standard $52$-card deck, what is $P(\text{ace or king})$ on one draw?

Answer: $\frac{2}{13}$. Eight cards total (4 aces + 4 kings) out of 52.

Flashcard 12: What is the multiplication rule for independent events?

Answer: $P(A \cap B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.

Flashcard 13: State the addition rule when events $A$ and $B$ are mutually exclusive.

Answer: $P(A\cup B)=P(A)+P(B)$. No overlap to subtract when events cannot occur together.

Flashcard 14: Identify the probability of rolling a 3 on a fair six-sided die.

Answer: $\frac{1}{6}$. One favorable outcome (rolling 3) out of 6 possible outcomes.

Flashcard 15: What is the probability of rolling a 2 or a 5 on a six-sided die?

Answer: $\frac{1}{3}$. Two favorable outcomes (2 or 5) out of 6 possible outcomes.

Flashcard 16: If $P(A\cap B)=0.12$ and $P(B)=0.3$, what is $P(A\mid B)$?

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

Flashcard 17: What is the definition of a mutually exclusive event?

Answer: Events that cannot occur at the same time. Two events that have no outcomes in common.

Flashcard 18: Define an event in probability.

Answer: A set of outcomes of a probability experiment. A subset of the sample space.

Flashcard 19: Identify the probability of not rolling a 5 on a six-sided die.

Answer: $\frac{5}{6}$. Complement of rolling a 5, which has probability $\frac{1}{6}$.

Flashcard 20: Identify the probability of flipping tails on a fair coin.

Answer: $\frac{1}{2}$. One of two equally likely outcomes on a fair coin.

Flashcard 21: 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 $B$ has occurred.

Flashcard 22: Define independent events.

Answer: Events are independent if the occurrence of one does not affect the other. One event's outcome doesn't influence the other.

Flashcard 23: State the probability of drawing a spade or a heart from a deck of cards.

Answer: $\frac{1}{2}$. 26 spades and hearts out of 52 total cards.

Flashcard 24: What is $P(\text{exactly one head})$ when tossing two fair coins?

Answer: $\frac{1}{2}$. Two outcomes HT and TH out of four total possibilities.

Flashcard 25: Find the probability of drawing a red card and then a black card without replacement.

Answer: $\frac{1}{2}$. $\frac{26}{52} \times \frac{26}{51}$ for dependent draws without replacement.

Flashcard 26: Identify the probability of drawing two Aces consecutively without replacement.

Answer: $\frac{1}{221}$. Without replacement: $\frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$.

Flashcard 27: Find the probability of rolling a number greater than 4 on a six-sided die.

Answer: $\frac{1}{3}$. Numbers 5 and 6 are greater than 4 out of 6 outcomes.

Flashcard 28: What is the probability of rolling a number less than 4 on a six-sided die?

Answer: $\frac{1}{2}$. Numbers 1, 2, and 3 are less than 4.

Flashcard 29: State the probability of flipping two heads in two coin flips.

Answer: $\frac{1}{4}$. Independent events: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

Flashcard 30: State the addition rule for mutually exclusive events.

Answer: $P(A \text{ or } B) = P(A) + P(B)$. Add probabilities when events cannot occur simultaneously.

Flashcard 31: What is the formula for the probability of complementary events?

Answer: $P(A') = 1 - P(A)$. The complement probability equals 1 minus the original probability.

Flashcard 32: What is the number of permutations of $5$ distinct items taken $2$ at a time?

Answer: $20$. Permutation formula: $\frac{5!}{(5-2)!} = \frac{5!}{3!} = 20$.

Flashcard 33: State the multiplication rule for independent events.

Answer: $P(A \text{ and } B) = P(A) \times P(B)$. Multiply probabilities when events don't affect each other.

Flashcard 34: What is the formula for the probability of complementary events?

Answer: $P(A') = 1 - P(A)$. The complement probability equals 1 minus the original probability.

Flashcard 35: 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 36: What condition indicates that events $A$ and $B$ are independent using conditional probability?

Answer: $P(A\mid B)=P(A)$. Knowledge of $B$ doesn't change probability of $A$ when independent.

Flashcard 37: What is the probability that an event will not occur?

Answer: 1 - P(A). The complement rule for any event A.

Flashcard 38: What is the probability of not drawing a face card from a standard deck?

Answer: $\frac{10}{13}$. 40 non-face cards out of 52 total cards.

Flashcard 39: A bag has $3$ red and $2$ blue marbles. What is $P(\text{blue})$ on one draw?

Answer: $\frac{2}{5}$. Two blue marbles out of five total marbles.

Flashcard 40: What is the probability of drawing a heart from a full deck of cards?

Answer: $\frac{1}{4}$. 13 hearts out of 52 cards in a standard deck.

Flashcard 41: What is the probability of drawing a black card from a standard deck?

Answer: $\frac{1}{2}$. 26 black cards (clubs and spades) out of 52 total cards.

Flashcard 42: Define a probability experiment.

Answer: An action or trial with uncertain outcomes. A process with multiple possible results.

Flashcard 43: State the probability of drawing an ace from a standard deck of cards.

Answer: $\frac{1}{13}$. 4 aces out of 52 total cards.

Flashcard 44: What is the addition rule for probability?

Answer: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Accounts for overlap when events can occur together.

Flashcard 45: Identify the probability of rolling a 3 on a fair six-sided die.

Answer: $\frac{1}{6}$. One favorable outcome (rolling 3) out of 6 possible outcomes.

Flashcard 46: Calculate the probability of rolling a 2 or a 5 on a six-sided die.

Answer: $\frac{1}{3}$. Two favorable outcomes (2 or 5) out of six possible.

Flashcard 47: What is the formula for permutations of $n$ distinct items taken $r$ at a time?

Answer: $\frac{n!}{(n-r)!}$. Order matters in permutations - arrangements of items.

Flashcard 48: What condition indicates that events $A$ and $B$ are mutually exclusive (disjoint)?

Answer: $P(A\cap B)=0$. Events cannot happen simultaneously when mutually exclusive.

Flashcard 49: What is the probability of rolling an even number on a six-sided die?

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

Flashcard 50: What is the valid range of any probability $P(E)$?

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

Flashcard 51: From a $52$-card deck, what is $P(\text{ace or heart})$ on one draw?

Answer: $\frac{4}{13}$. Use addition rule: $\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52}$.

Flashcard 52: Identify the probability of drawing two aces in a row from a deck, without replacement.

Answer: $\frac{1}{221}$. $\frac{4}{52} \times \frac{3}{51}$ for dependent events without replacement.

Flashcard 53: Identify the probability of picking a red marble from a bag of 3 red and 7 blue marbles.

Answer: $\frac{3}{10}$. Three red marbles out of ten total marbles.

Flashcard 54: What is the definition of independent events in probability?

Answer: Events where the outcome of one does not affect the other. One event's outcome doesn't influence the other's probability.

Flashcard 55: Calculate the probability of rolling an even number on a six-sided die.

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

Flashcard 56: What is the probability of flipping two heads in a row with a fair coin?

Answer: $\frac{1}{4}$. $\frac{1}{2} \times \frac{1}{2}$ for two independent coin flips.

Flashcard 57: A jar has $4$ green and $6$ yellow marbles. Without replacement, what is $P(\text{green then green})$?

Answer: $\frac{2}{15}$. First green: $\frac{4}{10}$, then green: $\frac{3}{9}$, multiply together.

Flashcard 58: What is the definition of independent events in probability?

Answer: Events where the outcome of one does not affect the other. One event's outcome doesn't influence the other's probability.

Flashcard 59: What is $P(\text{heads})$ for one fair coin toss?

Answer: $\frac{1}{2}$. One favorable outcome (heads) out of two total.

Flashcard 60: 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 61: What is the definition of a mutually exclusive event?

Answer: Events that cannot occur at the same time. Two events that have no outcomes in common.

Flashcard 62: Identify the probability of flipping tails on a fair coin.

Answer: $\frac{1}{2}$. One of two equally likely outcomes on a fair coin.

Flashcard 63: What is the probability of getting exactly $2$ heads in $3$ fair coin tosses?

Answer: $\frac{3}{8}$. Three ways (HHT, HTH, THH) out of eight total outcomes.

Flashcard 64: What is the number of combinations when choosing $2$ items from $5$ distinct items?

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

Flashcard 65: State the addition rule for any events $A$ and $B$ (not necessarily disjoint).

Answer: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Subtract overlap to avoid double-counting when events can occur together.

Flashcard 66: What is the probability of drawing a face card from a standard deck of cards?

Answer: $\frac{3}{13}$. 12 face cards (J, Q, K in each suit) out of 52 total cards.

Flashcard 67: A jar has $4$ green and $6$ yellow marbles. With replacement, what is $P(\text{green then green})$?

Answer: $\frac{4}{25}$. Each draw: $\frac{4}{10}$, multiply: $(\frac{4}{10})^2 = \frac{16}{100}$.

Flashcard 68: A jar has $4$ green and $6$ yellow marbles. With replacement, what is $P(\text{green then green})$?

Answer: $\frac{4}{25}$. Each draw: $\frac{4}{10}$, multiply: $(\frac{4}{10})^2 = \frac{16}{100}$.

Flashcard 69: What is the probability of drawing a card that is not a heart from a deck?

Answer: $\frac{3}{4}$. 39 non-heart cards out of 52 total cards.

Flashcard 70: What is the sample space of flipping a fair coin?

Answer: {Heads, Tails}. All possible outcomes of the experiment.

Flashcard 71: What condition indicates that events $A$ and $B$ are mutually exclusive (disjoint)?

Answer: $P(A\cap B)=0$. Events cannot happen simultaneously when mutually exclusive.

Flashcard 72: If $P(A)=0.7$, what is $P(A^c)$?

Answer: $0.3$. Complement rule: $1 - 0.7 = 0.3$.

Flashcard 73: What is the probability of drawing a black king from a standard deck?

Answer: $\frac{1}{26}$. 2 black kings out of 52 total cards.

Flashcard 74: What condition indicates that events $A$ and $B$ are independent using conditional probability?

Answer: $P(A\mid B)=P(A)$. Knowledge of $B$ doesn't change probability of $A$ when independent.

Flashcard 75: If $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

Answer: $0.7$. Addition rule: $0.4 + 0.5 - 0.2 = 0.7$.

Flashcard 76: From a $52$-card deck, what is $P(\text{ace or heart})$ on one draw?

Answer: $\frac{4}{13}$. Use addition rule: $\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52}$.

Flashcard 77: What is the probability of randomly choosing a vowel from the English alphabet?

Answer: $\frac{5}{26}$. Five vowels (A, E, I, O, U) out of 26 letters.

Flashcard 78: If $P(A\cap B)=0.12$ and $P(B)=0.3$, what is $P(A\mid B)$?

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

Flashcard 79: State the addition rule for mutually exclusive events.

Answer: $P(A \text{ or } B) = P(A) + P(B)$. Add probabilities when events cannot occur simultaneously.

Flashcard 80: What is $P(\text{at least one head})$ when tossing two fair coins?

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

Flashcard 81: What is the probability of drawing a card that is not a spade?

Answer: $\frac{3}{4}$. 39 non-spade cards out of 52 total cards.

Flashcard 82: State the addition rule for any events $A$ and $B$ (not necessarily disjoint).

Answer: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Subtract overlap to avoid double-counting when events can occur together.

Flashcard 83: What is the probability of rolling a 2 or a 5 on a six-sided die?

Answer: $\frac{1}{3}$. Two favorable outcomes (2 or 5) out of 6 possible outcomes.

Flashcard 84: Calculate the probability of drawing a Queen, replacing it, then drawing a King.

Answer: $\frac{1}{169}$. With replacement: $\frac{4}{52} \times \frac{4}{52} = \frac{1}{169}$.

Flashcard 85: What is $P(\text{sum }7)$ when rolling two fair six-sided dice?

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

Flashcard 86: If $P(A\cup B)=0.9$, what is $P((A\cup B)^c)$?

Answer: $0.1$. Complement rule: $1 - 0.9 = 0.1$.

Flashcard 87: What is the definition of probability for an event $E$ using equally likely outcomes?

Answer: $P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Basic definition using favorable outcomes over total outcomes.

Flashcard 88: If $P(A)=0.7$, what is $P(A^c)$?

Answer: $0.3$. Complement rule: $1 - 0.7 = 0.3$.

Flashcard 89: Find the probability of drawing a queen given the first card drawn was a king.

Answer: $\frac{4}{51}$. 4 queens remain out of 51 cards after drawing a king.

Flashcard 90: Identify the probability of rolling at least one 6 with two six-sided dice.

Answer: $\frac{11}{36}$. Use complement: $1 - P(\text{no 6's}) = 1 - \frac{25}{36}$.

Flashcard 91: What is $P(\text{sum }2)$ when rolling two fair six-sided dice?

Answer: $\frac{1}{36}$. Only one way to get sum 2: (1,1).

Flashcard 92: A spinner has $8$ equal sections numbered $1$ to $8$. What is $P(\text{multiple of }3)$?

Answer: $\frac{1}{4}$. Multiples of 3 are 3 and 6, so $\frac{2}{8} = \frac{1}{4}$.

Flashcard 93: From a standard $52$-card deck, what is $P(\text{ace})$ on one draw?

Answer: $\frac{1}{13}$. Four aces in a standard 52-card deck.

Flashcard 94: What is the counting principle for $k$ steps with $n_1,n_2,\dots,n_k$ choices?

Answer: $n_1\cdot n_2\cdots n_k$. Multiply choices at each step for total possibilities.

Flashcard 95: What is the probability formula for a single event?

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

Flashcard 96: What is the probability of drawing a face card from a standard deck?

Answer: $\frac{3}{13}$. 12 face cards (Jacks, Queens, Kings) out of 52 total cards.

Flashcard 97: A bag has $3$ red and $2$ blue marbles. What is $P(\text{blue})$ on one draw?

Answer: $\frac{2}{5}$. Two blue marbles out of five total marbles.

Flashcard 98: If $P(A)=0.3$ and $P(B)=0.5$ and $A,B$ are independent, what is $P(A\cap B)$?

Answer: $0.15$. Multiply probabilities for independent events: $0.3 \times 0.5$.

Flashcard 99: Calculate the probability of drawing a Queen, replacing it, then drawing a King.

Answer: $\frac{1}{169}$. With replacement: $\frac{4}{52} \times \frac{4}{52} = \frac{1}{169}$.

Flashcard 100: If $P(A)=0.6$ and $P(B)=0.2$ and $A,B$ are mutually exclusive, what is $P(A\cup B)$?

Answer: $0.8$. Add probabilities for mutually exclusive events: $0.6 + 0.2$.