Probability - ACT Math

1. What is the total probability formula for a sample space S?

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

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

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

3. What is the probability of the complement of $A\cup B$ in terms of $P(A\cup B)$?

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

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

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

5. What is the complement rule for an event $E$?

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

6. What is the probability of a certain event (an event that must happen)?

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

7. What is the multiplication rule for independent events?

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

8. Identify the probability of rolling a 3 on a fair six-sided die.

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

9. What is the probability of rolling a 2 or a 5 on a six-sided die?

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

10. If $P(A\cap B)=0.12$ and $P(B)=0.3$, what is $P(A\mid B)$?

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

11. What is the definition of a mutually exclusive event?

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

12. Define an event in probability.

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

13. Identify the probability of not rolling a 5 on a six-sided die.

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

14. Identify the probability of flipping tails on a fair coin.

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

15. What is the definition of conditional probability $P(A\mid B)$?

    $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. Probability of $A$ given $B$ has occurred.

16. Define independent events.

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

17. What is $P(\text{exactly one head})$ when tossing two fair coins?

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

18. Identify the probability of drawing two Aces consecutively without replacement.

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

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

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

20. What is the probability of rolling a number less than 4 on a six-sided die?

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

21. State the probability of flipping two heads in two coin flips.

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

22. State the addition rule for mutually exclusive events.

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

23. What is the formula for the probability of complementary events?

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

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

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

25. What is the probability of drawing a black card from a standard deck?

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

26. State the probability of drawing an ace from a standard deck of cards.

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

27. What is the addition rule for probability?

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

28. Calculate the probability of rolling a 2 or a 5 on a six-sided die.

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

29. What is the formula for permutations of $n$ distinct items taken $r$ at a time?

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

30. What is the probability of rolling an even number on a six-sided die?

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

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

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

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

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

33. What is the definition of independent events in probability?

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

34. Calculate the probability of rolling an even number on a six-sided die.

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

35. What is the probability of flipping two heads in a row with a fair coin?

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

36. What is $P(\text{heads})$ for one fair coin toss?

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

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

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

38. What is the probability of drawing a card that is not a heart from a deck?

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

39. What is the sample space of flipping a fair coin?

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

40. What condition indicates that events $A$ and $B$ are mutually exclusive (disjoint)?

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

41. If $P(A)=0.7$, what is $P(A^c)$?

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

42. What condition indicates that events $A$ and $B$ are independent using conditional probability?

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

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

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

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

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

45. What is the probability of drawing a card that is not a spade?

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

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

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

47. If $P(A\cup B)=0.9$, what is $P((A\cup B)^c)$?

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

48. What is the definition of probability for an event $E$ using equally likely outcomes?

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

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

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

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

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

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

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

52. Calculate the probability of drawing a Queen, replacing it, then drawing a King.

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

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

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

54. State the probability of an impossible event.

    0. Impossible events can never happen.

55. What is the valid range of any probability $P(E)$?

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

56. If $P(A)=0.2$ and $P(B)=0.6$ and $A,B$ are independent, what is $P(A\cup B)$?

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

57. Calculate the probability of not rolling a 5 on a six-sided die.

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

58. What is the sum of probabilities of all outcomes in a sample space?

    1. All possible outcomes must account for the entire sample space.

59. Define a probability experiment.

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

60. Identify the probability of rolling a 3 on a six-sided die.

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

61. Identify the probability of drawing a King on the first draw from a deck.

    $\frac{1}{13}$. Four Kings out of 52 cards in a standard deck.

62. What is the probability of getting exactly $2$ heads in $3$ fair coin tosses?

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

63. State the complementary rule for probability.

    P($A^c$) = 1 - P(A). The complement of event A has probability 1 minus P(A).

64. State the multiplication rule for independent events $A$ and $B$.

    $P(A\cap B)=P(A)P(B)$. Independent events don't affect each other's probabilities.

65. What is the formula for combinations of $n$ distinct items chosen $r$ at a time?

    $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Order doesn't matter in combinations - just selection.

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

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

67. What is the number of outcomes when rolling two fair six-sided dice (ordered pairs)?

    $36$. Each die has 6 outcomes, so $6 \times 6 = 36$ total.

68. If $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

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

69. What is the number of permutations of $5$ distinct items taken $2$ at a time?

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

70. What is the probability of drawing an Ace from a standard deck of cards?

    $\frac{1}{13}$. Four Aces out of 52 total cards in a standard deck.

71. From a standard $52$-card deck, what is $P(\text{heart})$ on one draw?

    $\frac{1}{4}$. Thirteen hearts in a standard 52-card deck.

72. What is the probability of randomly choosing a vowel from the English alphabet?

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

73. Identify the probability of selecting a prime number from numbers 1 to 10.

    $\frac{4}{10}$. Prime numbers from 1-10 are: 2, 3, 5, 7.

74. If $P(A)=0.3$ and $P(B)=0.5$ and $A,B$ are independent, what is $P(A\cap B)$?

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

75. State the formula for conditional probability.

    $P(A|B) = \frac{P(A \cap B)}{P(B)}$. Probability of A given that B has already occurred.

76. State the addition rule when events $A$ and $B$ are mutually exclusive.

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

77. A bag has $3$ red and $2$ blue marbles. What is $P(\text{not red})$ on one draw?

    $\frac{2}{5}$. Complement of red is blue, which has probability $\frac{2}{5}$.

78. What is the probability of drawing a black king from a standard deck?

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

79. What is the probability of selecting a consonant from the English alphabet?

    $\frac{21}{26}$. 21 consonants out of 26 letters in the alphabet.

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

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

81. Define complementary events.

    Two events are complementary if their probabilities sum to 1. One event is the complement of the other.

82. If $P(A)=0.25$ and $P(A\mid B)=0.25$, what relationship holds between $A$ and $B$?

    They are independent. When $P(A\mid B) = P(A)$, events are independent.

83. From a standard $52$-card deck, what is $P(\text{face card})$ on one draw?

    $\frac{3}{13}$. Twelve face cards (J, Q, K in each of 4 suits).

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

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

85. State the multiplication rule for independent events.

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

86. What is the probability of an impossible event?

    $0$. An event that cannot occur has probability 0.

87. Identify the probability of rolling a prime number on a six-sided die.

    $\frac{1}{2}$. Prime numbers 2, 3, 5 out of 6 possible outcomes.

88. What is the probability of drawing a heart from a full deck of cards?

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

89. State the general multiplication rule for events $A$ and $B$ using conditional probability.

    $P(A\cap B)=P(A)P(B\mid A)$. General rule using conditional probability for any events.

90. Find the probability of drawing a heart from a standard deck of cards.

    $\frac{1}{4}$. 13 hearts out of 52 total cards.

91. State the probability of a certain event.

    1. Certain events always happen.

92. Define mutually exclusive events.

    Events that cannot occur at the same time. No overlap between the events is possible.

93. What is the probability of drawing a red card or a club from a deck?

    $\frac{7}{13}$. 26 red cards plus 13 clubs minus 13 overlapping red clubs.

94. What is the probability of drawing a red card from a standard deck?

    $\frac{1}{2}$. 26 red cards (hearts and diamonds) out of 52 total cards.

95. What is the probability of rolling a sum of 7 with two six-sided dice?

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

96. What is the probability that an event will not occur?

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

97. What is the counting principle for $k$ steps with $n_1,n_2,\dots,n_k$ choices?

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

98. Find the probability of drawing a queen given the first card drawn was a king.

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

99. State the probability of drawing a spade or a heart from a deck of cards.

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

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

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

101. What is the probability of drawing a face card from a standard deck of cards?

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

102. Define conditional probability.

     The probability of an event given that another event has occurred. Probability changes based on prior information.

103. Find the probability of drawing a red card and then a black card without replacement.

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

104. What is the number of combinations when choosing $2$ items from $5$ distinct items?

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

105. What is the probability of flipping heads on a fair coin?

     $\frac{1}{2}$. One favorable outcome (heads) out of 2 possible outcomes.

106. What is the probability of not drawing a face card from a standard deck?

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

107. What is the probability formula for a single event?

     $Probability = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition for any single event.

108. What is the symmetry property of intersection for events $A$ and $B$?

     $P(A\cap B)=P(B\cap A)$. Intersection is commutative - order doesn't matter.

109. Identify the probability of rolling a sum of 7 with two six-sided dice.

     $\frac{1}{6}$. 6 ways to roll sum of 7 out of 36 total outcomes.

110. What is the probability of drawing a face card from a standard deck?

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

Study these flashcards