Probability - PSAT Math

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    If $A\cap B=\varnothing$, then $P(A\cup B)=P(A)+P(B)$. Disjoint events have no overlap, so just add their probabilities.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

76. 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

87. 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 equals their intersection divided by B's probability.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

101. What is the probability formula for an event $A$ using favorable and total outcomes?

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

102. What is the formula for the complement of an event $A$?

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

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

     $\frac{n!}{(n-r)!}$. Order matters; divide out the $(n-r)!$ unused items.

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

     $\frac{1}{2}$. Three prime numbers $(2,3,5)$ out of six equally likely outcomes.

105. If $P(A)=0.7$, $P(B)=0.4$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

     $0.9$. Using addition rule: $0.7+0.4-0.2=0.9$

106. What is $P(A\cup B)$ if $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$?

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

107. What is the definition of independence using probabilities of intersection?

     $A,B\text{ independent if }P(A\cap B)=P(A)P(B)$. Events are independent when their intersection equals the product of individual probabilities.

108. What is the general multiplication rule relating $P(A\cap B)$ and $P(A\mid B)$?

     $P(A\cap B)=P(B)P(A\mid B)$. Rearranges the conditional probability formula to find joint probability.

109. If $P(A)=0.25$ and $P(B)=0.6$ are independent, what is $P(A\cap B)$?

     $0.15$. Multiply independent probabilities: $0.25 \times 0.6 = 0.15$.

110. If $P(A)=0.37$, what is $P(A^c)$?

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

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

     $^nC_r=\frac{n!}{r!(n-r)!}$. Order doesn't matter: choose r items from n total items.

112. What is $P(A^c)$ if $P(A)=0.37$?

     $0.63$. Using complement rule: $1-0.37=0.63$.

113. What is $P(A\cup B)$ if $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.3$?

     $0.8$. General addition rule: $0.6 + 0.5 - 0.3 = 0.8$.

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

     $\frac{3}{10}$. $\frac{3}{5}\times\frac{2}{4}$ since one red is removed.

115. What is the multiplication rule for independent events $A$ and $B$?

     $P(A\cap B)=P(A)P(B)$. Independent events multiply: one doesn't affect the other.

116. What is the addition rule when events $A$ and $B$ are mutually exclusive?

     $P(A\cup B)=P(A)+P(B)$. When events can't occur together, simply add their probabilities.

117. What condition means events $A$ and $B$ are mutually exclusive?

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

118. What is $P(A\cap B)$ if $P(A)=0.2$ and $P(B)=0.5$ and $A$ and $B$ are independent?

     $0.1$. Independent events: multiply $0.2 \times 0.5 = 0.1$.

119. What is the probability of drawing a red card from a standard $52$-card deck?

     $\frac{1}{2}$. Half the deck (26 cards) are red.

120. Identify the probability of a certain event.

     $1$. Certain events always occur with probability one.

121. What is the probability of drawing $2$ red cards (without replacement) from a standard $52$-card deck?

     $\frac{25}{102}$. $\frac{26}{52} \times \frac{25}{51} = \frac{25}{102}$ for 2 reds without replacement.

122. What is the addition rule for any two events $A$ and $B$?

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

123. What is $P(A\cup B)$ if $A$ and $B$ are mutually exclusive, $P(A)=0.2$, and $P(B)=0.5$?

     $0.7$. Mutually exclusive events: $P(A\cup B) = 0.2 + 0.5 = 0.7$.

124. If $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$, what is $P(A\cup B)$?

     $0.9$. Use $P(A)+P(B)-P(A\cap B)=0.6+0.5-0.2$.

125. What is the probability formula for an event $A$ using equally likely outcomes?

     $P(A)=\frac{\text{favorable}}{\text{total}}$. Counts favorable outcomes divided by all possible outcomes.

126. What condition must hold for events $A$ and $B$ to be mutually exclusive?

     $P(A\cap B)=0$. Mutually exclusive events cannot occur simultaneously, so their intersection is empty.

127. How many outcomes are in a sequence of $3$ coin flips?

     $2^3=8$. Each flip has 2 outcomes; $2\times 2\times 2=8$.

128. Two fair dice are rolled. What is $P(\text{at least one }6)$?

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

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

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

130. Find $P(\text{exactly }2\text{ heads})$ when flipping $3$ fair coins.

     $\frac{3}{8}$. Three ways to get exactly $2$ heads (HHT, HTH, THH) out of $8$ total.

131. What is $P(\text{not }A)$ if $P(A)=0.37$?

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

132. What is $P(\text{at least one head})$ when a fair coin is flipped $3$ times?

     $\frac{7}{8}$. $1 - P(\text{all tails}) = 1 - \frac{1}{8} = \frac{7}{8}$.

133. What condition must hold for events $A$ and $B$ to be independent?

     $P(A\cap B)=P(A)P(B)$. Events are independent when their joint probability equals the product of individual probabilities.

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

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

135. What is the addition rule for mutually exclusive events $A$ and $B$?

     $P(A\cup B)=P(A)+P(B)$. For events that can't occur together, add their probabilities.

136. Identify the event described by "$A$ or $B$" using set notation.

     $A\cup B$. Union symbol represents "or" in probability.

137. Identify the sample space size when flipping $3$ fair coins.

     $2^3=8$. Each coin has $2$ outcomes; $3$ coins give $2^3$ total outcomes.

138. What is the formula for the probability of exactly $k$ successes in $n$ trials with success rate $p$?

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

139. What condition must hold for events $A$ and $B$ to be mutually exclusive?

     $P(A\cap B)=0$. Mutually exclusive events have no overlap.

140. What does it mean for events $A$ and $B$ to be independent?

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

141. What does it mean if events $A$ and $B$ are mutually exclusive in symbols?

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

142. What is $P(\text{even})$ when rolling one fair six-sided die?

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

143. What is the probability of an event when outcomes are equally likely?

     $P(E)=\frac{\text{favorable}}{\text{total}}$. Count favorable outcomes and divide by total possible outcomes.

144. If $P(A)=0.3$ and $P(B)=0.4$ are independent, what is $P(A\cap B)$?

     $0.12$. Multiply independent probabilities: $0.3\times 0.4$.

145. A bag has $3$ red and $2$ blue marbles. One is drawn. What is $P(\text{blue})$?

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

146. What is the expected value formula for outcomes $x_i$ with probabilities $p_i$?

     $E=\sum x_i p_i$. Multiply each outcome by its probability and sum the results.

147. What is $P(A\cup B)$ if $P(A)=0.4$, $P(B)=0.3$, and $A$ and $B$ are mutually exclusive?

     $0.7$. Mutually exclusive: simply add $0.4 + 0.3 = 0.7$.

148. A fair coin is flipped $3$ times. What is $P(\text{at least one head})$?

     $\frac{7}{8}$. Use complement: $1-P(\text{all tails})=1-\frac{1}{8}=\frac{7}{8}$.

149. What is the complement rule for an event $A$?

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

150. What is the probability of rolling a $6$ on a fair six-sided die?

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

151. Two fair coins are flipped. What is $P(\text{exactly one head})$?

     $\frac{1}{2}$. Outcomes HT and TH give exactly one head: $\frac{2}{4}=\frac{1}{2}$.

152. What is the multiplication rule for any two events $A$ and $B$?

     $P(A\cap B)=P(A)\cdot P(B\mid A)$. Multiply probability of A by probability of B given A occurred.

153. A bag has $5$ red and $3$ blue marbles. Two are drawn without replacement. What is $P(\text{both red})$?

     $\frac{5}{8}\cdot\frac{4}{7}=\frac{5}{14}$. First red is $\frac{5}{8}$, then $\frac{4}{7}$ remain; multiply for both red.

154. What is the formula for probability of an event $A$ using favorable and total outcomes?

     $P(A)=\frac{\text{favorable}}{\text{total}}$. Probability equals the ratio of favorable outcomes to total possible outcomes.

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

     $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$. The probability of $A$ given $B$ equals their intersection divided by $P(B)$.

156. A coin is flipped $2$ times. What is the probability of exactly one head?

     $\frac{1}{2}$. Two ways (HT, TH) out of four total outcomes (HH, HT, TH, TT).

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

     $0.7$. Apply addition rule: $0.5 + 0.4 - 0.2 = 0.7$.

158. What is the probability of the sample space $S$?

     $P(S)=1$. The entire sample space has probability 1 (something must happen).

159. What is $P(\text{not red})$ if $P(\text{red})=0.37$?

     $0.63$. Use complement rule: $1 - 0.37 = 0.63$.

160. Find $P(A\cup B)$ if $P(A)=0.50$, $P(B)=0.30$, and $P(A\cap B)=0.10$.

     $0.70$. Apply addition rule: $0.50+0.30-0.10=0.70$.

161. What is the formula for probability of an event $A$ using equally likely outcomes?

     $P(A)=\frac{\text{favorable}}{\text{total}}$. Divide the number of favorable outcomes by the total number of outcomes.

162. A spinner has $4$ equal sections numbered $1$ to $4$. What is $P(\text{odd})$?

     $\frac{1}{2}$. Two odd numbers (1 and 3) out of four equally likely outcomes.

163. 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.

164. What is $P(A\mid B)$ if $P(A\cap B)=0.12$ and $P(B)=0.3$?

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

165. What is $P(A\cup B)$ if $P(A)=0.4$, $P(B)=0.5$, and $A$ and $B$ are mutually exclusive?

     $0.9$. Mutually exclusive events: $0.4+0.5=0.9$.

166. What is the general addition rule for any events $A$ and $B$?

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

167. What is the probability range rule for any event $A$?

     $0\le P(A)\le 1$. All probabilities must be between 0 (impossible) and 1 (certain).

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

     $\frac{3}{8}$. 3 ways to get 2 heads (HHT, HTH, THH) out of 8 total outcomes.

169. What is the multiplication principle for $k$ stages with $n_1,n_2,\dots,n_k$ choices?

     $n_1\cdot n_2\cdot\dots\cdot n_k$. Multiply choices at each stage for total number of outcomes.

170. What is $P(A\cup B)$ when $A$ and $B$ are mutually exclusive?

     $P(A\cup B)=P(A)+P(B)$. No overlap means add probabilities without subtracting intersection.

171. Find $P(A\mid B)$ if $P(A\cap B)=0.18$ and $P(B)=0.30$.

     $0.60$. Apply conditional formula: $\frac{0.18}{0.30}=0.60$.

172. What is 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 adding probabilities.

173. Which statement correctly defines independence of events $A$ and $B$ using conditional probability?

     $A\text{ and }B\text{ are independent if }P(A\mid B)=P(A)$. Events are independent when knowing B doesn't change A's probability.

174. Find $P(A\cap B)$ if $P(A)=\frac{2}{5}$ and $P(B)=\frac{3}{10}$ and $A,B$ are independent.

     $\frac{3}{25}$. Multiply independent probabilities: $\frac{2}{5}\times\frac{3}{10}=\frac{6}{50}=\frac{3}{25}$.

175. What is the multiplication rule for probability using conditional probability?

     $P(A\cap B)=P(A)\,P(B\mid A)$. Multiply the probability of A by the probability of B given A.

176. What is $P(A^c)$ if $P(A)=0.62$?

     $0.38$. Use complement rule: $1-0.62=0.38$.

177. Identify the probability of an impossible event.

     $0$. Impossible events have zero probability.

178. What is the probability of exactly $2$ heads in $3$ fair coin flips?

     $\binom{3}{2}\left(\frac{1}{2}\right)^3=\frac{3}{8}$. Binomial: $\binom{3}{2}=3$ ways, each with probability $\frac{1}{8}$.

179. What is the addition rule for any events $A$ and $B$?

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

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

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

181. If $P(A\mid B)=0.2$ and $P(B)=0.5$, what is $P(A\cap B)$?

     $0.1$. Use multiplication rule: $0.2 \times 0.5 = 0.1$.

182. Find $P(A\cup B)$ if $A$ and $B$ are mutually exclusive, $P(A)=0.2$, and $P(B)=0.5$.

     $0.7$. For mutually exclusive events: $P(A \cup B) = 0.2 + 0.5 = 0.7$.

183. What is $P(A\cap B)$ if $A$ and $B$ are independent, $P(A)=0.4$, and $P(B)=0.25$?

     $0.10$. Multiply independent probabilities: $0.4×0.25=0.10$.

184. 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; divide by $r!$ to remove arrangements.

185. What is the total probability rule for an event $A$ and its complement $A^c$?

     $P(A)+P(A^c)=1$. An event and its complement together cover all possibilities, summing to $1$.

186. What condition means events $A$ and $B$ are independent in terms of conditional probability?

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

187. What is $P(A\cup B)$ if $P(A)=0.6$, $P(B)=0.5$, and $P(A\cap B)=0.2$?

     $0.9$. General addition: $0.6+0.5-0.2=0.9$.

188. What is the probability range for any event $A$?

     $0\le P(A)\le 1$. All probabilities must be between 0 and 1 inclusive.

Study these flashcards