Probability Basic Flashcards

1
Q

Define probability of an event E in simplest terms.

A

Probability = (Number of favorable outcomes) / (Total possible outcomes).

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2
Q

What is the range of any probability value?

A

From 0 to 1, inclusive.

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3
Q

If an event is certain to happen, its probability is…?

A

1.

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4
Q

If an event is impossible, its probability is…?

A

0.

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5
Q

Define sample space S.

A

The set of all possible outcomes of a probabilistic experiment.

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6
Q

If we flip one fair coin, what is P(Heads)?

A

1/2.

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7
Q

If we flip a fair coin once, P(Tails) =?

A

1/2, same reasoning as heads.

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8
Q

In a fair six-sided die roll, P(rolling a 3) =?

A

1/6.

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9
Q

In a fair six-sided die roll, P(rolling an odd number) =?

A

3/6 = 1/2 (odd faces are 1,3,5).

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10
Q

Define complement of an event A, denoted Aᶜ.

A

All outcomes in the sample space not in A; P(Aᶜ)=1−P(A).

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11
Q

If P(A)=0.3, what is P(Aᶜ)?

A

0.7.

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12
Q

If two events A and B are mutually exclusive (disjoint), P(A or B) =?

A

P(A)+P(B).

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13
Q

State the general addition rule for events A, B (not necessarily disjoint).

A

P(A or B)=P(A)+P(B)−P(A and B).

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14
Q

Two fair coin flips: sample space size?

A

4 outcomes: (HH, HT, TH, TT).

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15
Q

Two fair coin flips: P(exactly one head)?

A

2/4=1/2 (the outcomes are HT, TH).

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16
Q

If P(A)=0.4, P(B)=0.3, A, B disjoint => P(A or B)=?

A

0.4+0.3=0.7 (since disjoint means no overlap).

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17
Q

In rolling one fair die, P(multiple of 3)?

A

Faces multiple of 3: 3,6 => 2 out of 6 => 1/3.

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18
Q

In picking a random day of the week, P(weekend day)? (Assume Sat & Sun only.)

A

2/7.

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19
Q

If P(A)=0.25, P(B)=0.4, and A, B are disjoint, P(A and B) =?

A

0 (disjoint => cannot occur together).

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20
Q

Define conditional probability P(A|B).

A

Probability A occurs given B has occurred: P(A|B)=P(A and B)/P(B).

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21
Q

If P(A)=0.5, P(B)=0.4, P(A and B)=0.2 => P(A|B)=?

A

P(A|B)=0.2/0.4=0.5.

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22
Q

Events A, B are independent if P(A and B)=?

A

P(A)·P(B).

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23
Q

If A, B independent, then P(A|B)=?

A

P(A) (B’s occurrence doesn’t affect A).

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24
Q

Flip a fair coin 3 times: sample space size?

A

2³=8 outcomes.

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25
Q

Flip 3 fair coins: P(exactly 2 heads)?

A

Number ways=3 (HHT, HTH, THH). So 3/8.

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26
Q

Roll 2 fair dice: total outcomes =?

A

6×6=36.

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27
Q

Roll 2 fair dice: P(sum=7)?

A

Number of ways=6 (1+6,2+5,3+4,4+3,5+2,6+1), so 6/36=1/6.

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28
Q

Roll 2 fair dice: P(sum=12)?

A

Only (6,6) => 1/36.

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29
Q

Define random variable X in probability.

A

A numerical outcome assigned to each element in the sample space.

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30
Q

Bernoulli trial: P(success)=p => P(X=1)=? P(X=0)=?

A

P(X=1)=p, P(X=0)=1−p.

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31
Q

Binomial distribution: P(X=k) formula for n trials, success prob p?

A

P(X=k)= (nCk) p^k (1−p)^(n−k).

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32
Q

Mean of a Binomial(n,p) random variable?

A

μ= n p.

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33
Q

Variance of a Binomial(n,p) random variable?

A

σ²= n p (1−p).

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34
Q

Define uniform probability distribution on {1,2,…,n}.

A

Each outcome has probability 1/n.

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35
Q

If X~Binomial(4,0.5), P(X=2)=?

A

(4C2)(0.5)²(0.5)²= 6×(0.25)×(0.25)= 6×0.0625=0.375.

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36
Q

A fair coin (p=0.5) tossed 4 times => total heads. Probability of exactly 3 heads?

A

(4C3)(0.5)³(0.5)¹=4×0.125×0.5=0.25.

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37
Q

Poisson distribution formula for P(X=k), parameter λ>0?

A

P(X=k)= (λ^k e^(−λ))/ k!.

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38
Q

Mean of a Poisson(λ) random variable is…?

A

λ (also variance = λ).

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39
Q

Conditional probability formula again: P(A|B)=?

A

P(A and B)/P(B).

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40
Q

Define complement rule in probability: P(Aᶜ)=?

A

1 − P(A).

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41
Q

If event A is ‘die shows prime number’, prime faces are {2,3,5}, P(A)=?

A

3/6=1/2.

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42
Q

Odds in favor of event A is ratio P(A)/P(Aᶜ). If P(A)=1/3 => odds=?

A

1/3 : 2/3=1:2.

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43
Q

Law of Total Probability: P(B)= Σ P(B|Aᵢ)P(Aᵢ). T/F?

A

True, summing over partition events Aᵢ.

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44
Q

If P(A)=0.3, P(B)=0.3, P(A and B)=0.09 => Are A, B independent?

A

Check 0.3×0.3=0.09 => yes, they match => independent.

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45
Q

Bayes’ Theorem short form: P(A|B)=?

A

[P(B|A) P(A)] / P(B).

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46
Q

Probability that a random digit (0–9) is even?

A

5 even digits (0,2,4,6,8) => 5/10=1/2.

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47
Q

2 cards drawn from standard 52 deck (without replacement). P(both aces)?

A

(4/52)(3/51)= (1/13)(3/51)=3/663=1/221.

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48
Q

A single draw from deck: P(face card) if face=Jack,Queen,King => 12 face cards. Probability?

A

12/52=3/13.

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49
Q

If events are collectively exhaustive, their probabilities sum to…?

A

1.

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50
Q

If P(A and B)=0 => A, B are disjoint => P(A or B)=?

A

P(A)+P(B).

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51
Q

Dice: sum≥10 => sums 10,11,12. Number of combos? 10 => 3 combos, 11 =>2 combos, 12 =>1 combo => total 6. Probability=?

A

6/36=1/6.

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52
Q

Random coin flips 2 times: Probability 0 heads =>?

A

(1/2)×(1/2)=1/4 => TT outcome.

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53
Q

Random coin flips 2 times: Probability 2 heads =>?

A

1/4 => HH only.

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54
Q

Random coin flips 2 times: Probability at least 1 head =>?

A

3/4 => HH,HT,TH.

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55
Q

If P(E)=0.2 => odds against E is…?

A

0.8 : 0.2=4:1.

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56
Q

If P(A|B)=0.7 => interpret?

A

Given B occurred, chance of A is 0.7.

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57
Q

If X is a discrete random variable, sum of P(X=x) over all x=?

A

1.

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58
Q

Expected value E(X) formula for discrete X=?

A

Σ x·P(X=x).

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59
Q

Var(X)= E(X²) − [E(X)]² => T/F?

A

True, basic variance formula.

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60
Q

In a fair coin, P(H)=P(T)=0.5 => expected number of heads in 1 toss=?

A

0.5.

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61
Q

If we do 10 coin flips, number of heads ~ Binomial(10,0.5). Probability of 5 heads=?

A

(10C5)(0.5)^(10)= 252×(1/1024)= 252/1024 ≈ 0.246.

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62
Q

Geometric distribution’s first success at trial k => formula?

A

P(X=k)= (1−p)^(k−1) p (for k=1,2,…).

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63
Q

In rolling one fair die, P(prime face)? (2,3,5) => 3 outcomes => P=?

A

3/6=1/2.

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64
Q

Choose 2 from 4 objects. # ways=?

A

(4C2)=6 => if equally likely, each pair has prob 1/6.

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65
Q

2 children in a family. Probability both are girls (assuming boy/girl equally likely)?

A

1/4 => outcomes: (G,G),(G,B),(B,G),(B,B).

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66
Q

Deck of 52, probability of drawing spade?

A

13/52=1/4.

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67
Q

Deck of 52, probability card is spade or heart?

A

26/52=1/2 => half the deck.

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68
Q

If P(A)=0.7, P(B)=0.6, P(A and B)=0.3 => P(A or B)=?

A

0.7+0.6−0.3=1.0.

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69
Q

From 3 coin flips (8 outcomes), P(exactly 1 head) =>?

A

3 out of 8 =>3/8=0.375.

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70
Q

Spin a fair spinner with 4 equal sectors. Probability of landing on sector #3 =>?

A

1/4.

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71
Q

Toss 2 fair coins => sample space=4. Probability 2 heads =>?

A

1/4 => only (H,H).

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72
Q

Random integer from 1 to 100 => probability multiple of 10 =>?

A

10 multiples => 10/100=1/10.

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73
Q

Random integer from 1 to 100 => P(multiple of 3 or 5) using inclusion-exclusion => # of multiples 3=33, 5=20, 15=6 => total=?

A

33+20−6=47 => P=47/100.

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74
Q

An event has P=0.45 => interpret odds in favor => 0.45:0.55 => ~0.82:1 => T/F?

A

Yes, that’s correct ratio.

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75
Q

Roll one fair die => P(≥4) => outcomes {4,5,6} => 3 out of 6 =>?

A

1/2.

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76
Q

Probability that a random letter from English alphabet (26 letters) is vowel if vowels are {A,E,I,O,U}? => 5 out of 26 =>?

A

5/26.

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77
Q

2 events A,B independent => P(A and B)= 0.12 => P(A)=0.3 => P(B)=?

A

0.4 => because 0.3×B=0.12 => B=0.4.

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78
Q

A fair coin tossed 4 times => total outcomes 16 => P(≥3 heads) => 3 heads or 4 heads => (4C3)(1/16)+(4C4)(1/16)=4/16+1/16=5/16 => T/F?

A

True => 5/16=0.3125.

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79
Q

In a group of 10, pick 3 for committee => # ways=(10C3)=120 => Probability of a specific group of 3 is 1/120 => T/F?

A

True.

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80
Q

If P(A|B)= 0.3 => interpret as: when B occurs, A occurs 30% of the time => T/F?

A

True.

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81
Q

For Poisson(λ=2), P(X=0)= e^(−2)(2^0)/0!= e^(−2) => T/F?

A

True => about 0.1353.

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82
Q

Mean of Binomial(10,0.3)= n p => 3 => T/F?

A

Yes => 3.

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83
Q

Variance of Binomial(10,0.3)= n p(1−p)= 10×0.3×0.7= 2.1 => T/F?

A

True.

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84
Q

Dice sum problem: P(sum=2) => only (1,1) => 1/36 => T/F?

A

Yes => correct.

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85
Q

Dice sum problem: P(sum=3) => (1,2),(2,1)=2/36 => T/F?

A

True => 1/18.

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86
Q

Card deck: Probability of red card => half the deck => 26/52=1/2 => T/F?

A

True.

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87
Q

Card deck: Probability of drawing a face card => J,Q,K => 4 each suit => 12 total => 12/52=3/13 => T/F?

A

True.

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88
Q

If (1,2),(2,1)=2/36 => T/F?

A

True => 1/18.

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89
Q

If random integer from 1 to 4 => sample space {1,2,3,4} => Probability of prime => primes {2,3} => 2/4=1/2 => T/F?

A

True.

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90
Q

If P(A)=0.6, P(B)=0.7, P(A and B)=0.42 => Are they independent? => Check 0.6×0.7=0.42 => T/F?

A

True => they are independent.

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91
Q

Saying probability of an event is 0.2 => means 20% chance => T/F?

A

True.

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92
Q

Conditional formula again: P(A and B)= P(A|B)×P(B) => T/F?

A

True.

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93
Q

Pick one letter from ‘MATH’ => 4 letters => P(A)= Probability letter is ‘A’ => 1/4 => T/F?

A

True.

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94
Q

Probability of picking a random number from 1 to 10 that is multiple of 2 => 5 out of 10 => T/F?

A

True => 1/2.

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95
Q

In 3 coin flips, P(TTT)= (1/2)³=1/8 => T/F?

A

True.

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96
Q

In 3 coin flips, P(no heads)= 1/8 => T/F?

A

True => same scenario as TTT.

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97
Q

If two events are exhaustive => P(A or B)=1 => T/F?

A

True => they cover entire sample space.

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98
Q

If P(A)=0 => event A is impossible => T/F?

A

True.

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99
Q

Pick random digit 0–9 => Probability of <5 => digits {0,1,2,3,4} => 5 out of 10 => T/F?

A

True => 1/2.

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100
Q

Random day from week => Probability day is Monday => 1/7 => T/F?

A

True, equally likely assumption.

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101
Q

If we want P(A or B or C) for 3 events => we can use inclusion-exclusion => T/F?

A

True => P(A)+P(B)+P(C)−(pairs) + P(A and B and C).

102
Q

What is the formula for the probability of the complement of event A?

A

P(Aᶜ)=1−P(A).

103
Q

State the formula for the union of two events A and B (general addition rule).

A

P(A ∪ B)=P(A)+P(B)−P(A ∩ B).

104
Q

Formula for A, B disjoint: P(A ∪ B)=?

A

P(A)+P(B), because P(A ∩ B)=0.

105
Q

Define conditional probability P(A|B).

A

P(A|B)= P(A ∩ B)/P(B), if P(B)≠0.

106
Q

When are events A and B independent in terms of probabilities?

A

P(A ∩ B)= P(A)·P(B).

107
Q

Formula for conditional probability if A, B independent?

A

P(A|B)= P(A).

108
Q

Law of Total Probability formula for P(B) if {Aᵢ} partition the sample space?

A

P(B)= Σ P(B|Aᵢ)P(Aᵢ).

109
Q

Bayes’ Theorem formula in short?

A

P(A|B)= [P(B|A) P(A)] / P(B).

110
Q

Odds in favor of A in terms of P(A)?

A

Odds(A)= P(A) : [1−P(A)].

111
Q

Odds against A if P(A)=p =>?

A

(1−p) : p.

112
Q

Number of ways to choose k items from n: formula for nCk?

A

(nCk)= n! / [k! (n−k)!].

113
Q

Define the sample space size for n coin flips.

114
Q

Probability of exactly k heads in n fair coin flips?

A

(nCk)(1/2)ⁿ.

115
Q

Formula for the binomial distribution P(X=k) with parameters n, p?

A

P(X=k)= (nCk) p^k (1−p)^(n−k).

116
Q

Mean of a binomial random variable X ~ Binomial(n,p)?

117
Q

Variance of a binomial X ~ Binomial(n,p)?

A

Var(X)= n p (1−p).

118
Q

Geometric distribution: P(X=k) formula for first success on kth trial (p=success prob)?

A

P(X=k)= (1−p)^(k−1) p, (k=1,2,…).

119
Q

Mean of geometric distribution with parameter p?

120
Q

Poisson distribution formula for P(X=k) with rate λ>0?

A

P(X=k)= (λ^k e^(−λ))/k!.

121
Q

Mean (and also variance) of Poisson(λ) distribution?

122
Q

Empirical probability definition formula (based on trials)?

A

P(A)= (Number of times A occurs)/(Total trials).

123
Q

Complement rule formula again: P(Aᶜ)=?

124
Q

Conditional probability rearranged: P(A ∩ B)=?

A

P(A|B) P(B)= P(B|A) P(A).

125
Q

If random variable X has outcomes xᵢ, with probabilities pᵢ, sum pᵢ=1 => mean formula?

A

E[X]= Σ xᵢ pᵢ.

126
Q

Variance formula for discrete X: Var(X)=?

A

Σ (xᵢ² pᵢ) − [E(X)]².

127
Q

If events A and B are independent => formula for P(A or B)?

A

P(A)+P(B)−P(A)P(B).

128
Q

General multiplication rule: P(A ∩ B)=?

A

P(A) P(B|A)= P(B) P(A|B).

129
Q

Uniform probability distribution on n equally likely outcomes => probability of each outcome=?

130
Q

Permutation formula for n distinct items taken k at a time?

A

P(n,k)= n! / (n−k)!.

131
Q

Combination formula for n distinct items choose k => (nCk) again?

A

n! / [k!(n−k)!].

132
Q

Hypergeometric distribution formula concept: Probability of k successes from K available in N, drawing n?

A

P(X=k)= [(KCk)((N−K)C(n−k))]/ (NCn).

133
Q

Bernoulli random variable X with success prob p => P(X=1)=p => formula for mean, variance?

A

E[X]=p, Var(X)=p(1−p).

134
Q

If events A₁,…,Aₙ partition sample space => P(B)=?

A

Σ P(B|Aᵢ) P(Aᵢ). (Law of total probability).

135
Q

If X~Binomial(n,p), Probability P(X=0)=?

A

(nC0) p^0 (1−p)ⁿ= (1−p)ⁿ.

136
Q

If X~Poisson(λ), Probability P(X=0)=?

A

e^(−λ).

137
Q

In random experiment, sum of probabilities of all disjoint events covering S=?

138
Q

Continuous uniform distribution on [a,b], pdf f(x)=?

A

1/(b−a) for x in [a,b].

139
Q

Mean of continuous uniform [a,b] is?

140
Q

If X~Exp(λ), pdf is λ e^(−λx). Probability formula for X≥0 => T/F?

A

True => That’s exponential distribution.

141
Q

Exponential(λ) distribution mean=?

142
Q

Conditional probability formula for P(B|A)= P(B ∩ A)/P(A). T/F?

143
Q

Bayes formula: P(A|B)= [P(B|A)P(A)] / P(B). T/F?

144
Q

When do we say A and B are mutually exclusive?

A

If P(A ∩ B)=0.

145
Q

When do we say A and B are collectively exhaustive?

A

If A ∪ B covers entire sample space => P(A∪B)=1.

146
Q

If sample space S has N equally likely outcomes => P(A)= #A / N => T/F?

147
Q

Define joint probability of events (A,B).

A

P(A∩B)= Probability that both events occur.

148
Q

Define marginal probability from joint distribution: P(A)= Σ or ∫ over B => T/F?

149
Q

Conditional pmf for discrete: p(x|y)= p(x,y)/ p(y)? T/F?

150
Q

In random selection of 2 from n, # ways is (nC2). Probability formula can use that T/F?

151
Q

Odds in favor of event E is P(E)/P(Eᶜ). T/F?

152
Q

Flip a fair coin twice. What’s the probability of 2 heads?

A

Outcomes: HH, HT, TH, TT ⇒ 4 total ⇒ only HH is success ⇒ 1/4.

153
Q

Flip a fair coin twice. Probability of at least 1 head?

A

3 out of 4 ⇒ 3/4.

154
Q

Roll one fair six-sided die. Probability of rolling a 5 or 6?

155
Q

Roll two fair dice. Probability that sum=2?

A

Only (1,1) ⇒ 1/36.

156
Q

Roll two fair dice. Probability that sum=7?

A

6 outcomes => 6/36=1/6.

157
Q

Random integer from 1 to 10. Probability it’s even?

A

Even: {2,4,6,8,10} => 5 out of 10 => 1/2.

158
Q

Random integer from 1 to 10. Probability it’s prime?

A

Primes= {2,3,5,7} => 4 out of 10 => 2/5.

159
Q

Deck of 52 cards. Probability of drawing a heart?

A

13 hearts => 13/52=1/4.

160
Q

Deck of 52 cards. Probability card is a heart or spade?

A

26/52=1/2.

161
Q

Deck of 52, Probability card is an Ace?

A

4 out of 52 => 1/13.

162
Q

Drawing one card: Probability card is red face card?

A

6 out of 52 => 6/52=3/26.

163
Q

2 cards drawn from 52 without replacement. Probability both are Aces?

A

(4/52)(3/51)= (1/13)(3/51)= 3/663=1/221.

164
Q

2 fair coins. Probability exactly 1 head?

A

HT or TH => 2 out of 4 => 1/2.

165
Q

Pick random day from 7-day week. Probability of it being Monday?

166
Q

Choose 2 from 5 distinct items at random. Probability the pair includes a specific item X?

A

Ways total= (5C2)=10. Pairs including X => choose 1 from remaining 4 => (4C1)=4 => Probability=4/10=2/5.

167
Q

Spin a fair spinner with 5 equal sectors. Probability it lands on sector #3?

168
Q

Two fair dice rolled. Probability that sum≥10?

A

6 outcomes => 6/36=1/6.

169
Q

Pick a random digit 0–9. Probability digit is at least 5?

170
Q

Random integer 1–20. Probability multiple of 4?

A

Yes => 1/4.

171
Q

Roll one fair die. Probability prime face?

172
Q

One fair coin, toss 1 time. Probability heads=?

173
Q

Random letter from ‘CAT’. Probability letter is ‘A’?

174
Q

Pick 1 from {A,B,C,D} equally likely. Probability of picking C?

175
Q

2 coin flips: Probability 2 tails?

176
Q

Random integer from 1–6. Probability ≤2?

177
Q

Deck of 52, Probability that drawn card is black King?

178
Q

2 draws from deck without replacement. Probability 2 spades?

179
Q

Probability that random chord of a circle is > radius leads to paradoxical results. This is known as the Bertrand paradox?

180
Q

Pick random day from {Mon,Tue,Wed,Thu,Fri,Sat,Sun}. Probability weekend?

181
Q

A single coin toss: Probability not heads?

182
Q

Spin fair spinner with 8 equal regions. Probability region #7?

183
Q

Roll fair six-sided die. Probability (even number)?

184
Q

Binomial example: 3 coin flips. Probability exactly 2 heads?

185
Q

Pick random integer from 1–4. Probability prime?

186
Q

2 coin flips. Probability at least 1 tail?

187
Q

Flip fair coin 3 times. Probability all heads?

188
Q

Draw 1 from {apple, banana, cherry}, each equally likely. Probability banana?

189
Q

Deck 52. Probability is a heart or diamond?

190
Q

Random digit 0–9. Probability digit ≥8?

191
Q

Random letter from ‘HELLO’: Probability letter=‘L’?

192
Q

Dice sum problem: Probability sum=6?

193
Q

Dice sum=8?

194
Q

Probability of 3 heads in 4 tosses?

195
Q

Pick random integer from 1–12. Probability multiple of 3?

196
Q

Random choice from {red,blue,green}. Probability is green?

197
Q

2 draws from 52 deck w/o replacement. Probability 1st is spade, 2nd is heart?

198
Q

One fair coin. P(Heads)?

199
Q

1 draw from deck. Probability a black ace?

200
Q

One 6-sided die. Probability face≥5?

201
Q

Pick 1 from {0,1}. Probability pick 1?

202
Q

Flip a fair coin 3 times. Probability exactly 1 head?

A

Number ways = 3 (HTT, THT, TTH), total outcomes=8 ⇒ 3/8.

203
Q

Flip a fair coin 3 times. Probability at least 1 tail?

A

Complement is no tails => all heads => 1/8 ⇒ so 1−1/8=7/8.

204
Q

Roll one fair six-sided die. Probability of rolling a 1 or 2?

A

2 out of 6 ⇒ 1/3.

205
Q

Roll two fair dice. Probability that sum=4?

A

Sum=4 combos: (1,3),(2,2),(3,1) ⇒ 3 combos ⇒ 3/36=1/12.

206
Q

Random integer from 1 to 5. Probability equals 3 => ?

A

1 out of 5 ⇒ 1/5.

207
Q

Pick a random digit 0–9. Probability even?

A

Even digits = {0,2,4,6,8} => 5 out of 10 ⇒ 1/2.

208
Q

Pick a random digit 0–9. Probability prime (2,3,5,7)?

A

4 out of 10 ⇒ 2/5.

209
Q

Deck of 52. Probability spade or heart?

A

26 out of 52 ⇒ 1/2 (all red hearts + black spades?). Wait carefully: hearts are 13 red, spades are 13 black. Sum=26. Yes => 1/2.

210
Q

Deck of 52. Probability face card? (J,Q,K => 4 each suit =>12 total)

A

12/52=3/13.

211
Q

Random letter from the word ‘GEEK’ => 4 letters (G,E,E,K). Probability letter= E =>?

A

2 out of 4 => 1/2.

212
Q

Toss 2 fair coins: Probability 2 tails =>?

A

(1/2)×(1/2)=1/4 => TT outcome.

213
Q

If an event has P(A)=0.7 => P(not A)=?

A

0.3 => complement rule.

214
Q

Two events A,B disjoint => P(A or B)= P(A)+P(B). T/F?

A

True => no overlap.

215
Q

Sum of probabilities of all possible outcomes in a discrete sample space => ?

216
Q

Probability an odd face from one fair die => {1,3,5} => 3/6=1/2 => T/F?

217
Q

Two fair dice, Probability sum=3 => combos (1,2),(2,1) => 2/36=1/18 => T/F?

218
Q

Select random day of 7. Probability it’s Wed =>?

219
Q

Random letter from ‘MISS’: 4 letters => M,I,S,S => Probability letter= S => 2/4=1/2 => T/F?

220
Q

Pick integer from 1 to 6. Probability prime => {2,3,5} => 3/6=1/2 => T/F?

221
Q

Flip fair coin 4 times => total outcomes=2⁴=16 => Probability all tails => 1/16 => T/F?

222
Q

3 coin flips: Probability exactly 2 tails => (3C2)(1/2)³= 3×(1/8)=3/8 => T/F?

223
Q

Deck of 52: Probability of a Joker =>?

A

Typically 0, since standard deck has no Joker (or assume 52 excludes them).

224
Q

Random integer from 1 to 100 => Probability multiple of 10 => multiples are 10,20,…,100 => 10 out of 100 =>1/10 => T/F?

225
Q

Roll 2 dice => Probability sum≥11 => sums=11 or12 => combos => sum=11 =>(5,6),(6,5)=2 combos, sum=12 =>(6,6)=1 => total=3 => 3/36 =>1/12 => T/F?

226
Q

Pick random letter from ‘ABC’. Probability letter= ‘C’? =>1/3 => T/F?

227
Q

Random coin toss: Probability heads =>1/2 => T/F?

A

True => fair coin.

228
Q

2 draws from deck without replacement => Probability 2 hearts => (13/52)(12/51)= 13×12/(52×51)=> T/F?

A

Yes => that’s correct.

229
Q

Sample space with n equally likely outcomes => Probability of an event A is #(A)/n => T/F?

230
Q

When flipping 1 fair coin, P(H)=0.7 => is that correct?

A

No, for a fair coin P(H)=0.5 => 0.7 would be unfair.

231
Q

Pick integer 1–8 => Probability is >5 => outcomes {6,7,8} =>3/8 => T/F?

A

True => 3 out of 8.

232
Q

Dice problem: Probability sum=8 => combos (2,6),(3,5),(4,4),(5,3),(6,2)=5 => 5/36 => T/F?

233
Q

If P(A)=0.3, P(B)=0.5, A,B independent => P(A∩B)= (0.3)(0.5)=0.15 => T/F?

A

True => that’s the definition.

234
Q

Probability of picking red card from deck => 26 red cards => 26/52=1/2 => T/F?

235
Q

Probability of picking the Queen of hearts => 1/52 => T/F?

236
Q

Random choice from {red,green,blue} => Probability ‘blue’ =>1/3 => T/F?

A

Yes => 1/3.

237
Q

If event E has P(E)=0 => E is impossible => T/F?

238
Q

Two disjoint events A,B => P(A∩B)=0 => T/F?

239
Q

In a single die roll, Probability #≥5 => {5,6}=2 out of 6 =>1/3 => T/F?

240
Q

Pick integer from 1–12 => Probability multiple of 4 => {4,8,12}=3 => 3/12=1/4 => T/F?

A

Yes => 1/4.

241
Q

Probability ‘2 heads in 2 coin flips’ =>1/4 => T/F?

A

True => HH only.

242
Q

Probability exactly 3 heads in 3 coin flips =>(3C3)(1/2)³=1×1/8=1/8 => T/F?

A

True => HHH.

243
Q

Pick from 1–4 => Probability prime => {2,3}=2/4=1/2 => T/F?

244
Q

Deck of 52 => Probability card is King => 4/52=1/13 => T/F?

245
Q

In drawing 1 from {a,b,c,d}, Probability (not c)=> 3/4 => T/F?

A

True => complement approach.

246
Q

Roll 2 dice => Probability sum=9 => combos= (3,6),(4,5),(5,4),(6,3)=4 => 4/36=1/9 => T/F?

247
Q

If P(A)=0.6 => P(Aᶜ)= 0.4 => T/F?

A

True => complement rule.

248
Q

Probability event A occurs at least once in 2 trials =>1− P(A never occurs) => T/F?

A

Yes => correct approach.

249
Q

A fair coin tossed once: P(Tails)= 1/2 => T/F?

250
Q

Pick random integer from 1–100 => Probability it’s 1 => 1/100 => T/F?

A

Yes => 1 out of 100.

251
Q

Pick random letter from ‘MATH’ => Probability letter ‘T’? => {M,A,T,H} => T =>1/4 => T/F?

A

True => equally likely.