Probability Basic Flashcards
1
Q
Define probability of an event E in simplest terms.
A
Probability = (Number of favorable outcomes) / (Total possible outcomes).
2
Q
What is the range of any probability value?
A
From 0 to 1, inclusive.
3
Q
If an event is certain to happen, its probability is…?
A
1.
4
Q
If an event is impossible, its probability is…?
A
0.
5
Q
Define sample space S.
A
The set of all possible outcomes of a probabilistic experiment.
6
Q
If we flip one fair coin, what is P(Heads)?
A
1/2.
7
Q
If we flip a fair coin once, P(Tails) =?
A
1/2, same reasoning as heads.
8
Q
In a fair six-sided die roll, P(rolling a 3) =?
A
1/6.
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).
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).
11
Q
If P(A)=0.3, what is P(Aᶜ)?
A
0.7.
12
Q
If two events A and B are mutually exclusive (disjoint), P(A or B) =?
A
P(A)+P(B).
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).
14
Q
Two fair coin flips: sample space size?
A
4 outcomes: (HH, HT, TH, TT).
15
Q
Two fair coin flips: P(exactly one head)?
A
2/4=1/2 (the outcomes are HT, TH).
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).
17
Q
In rolling one fair die, P(multiple of 3)?
A
Faces multiple of 3: 3,6 => 2 out of 6 => 1/3.
18
Q
In picking a random day of the week, P(weekend day)? (Assume Sat & Sun only.)
A
2/7.
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).
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).
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.
22
Q
Events A, B are independent if P(A and B)=?
A
P(A)·P(B).
23
Q
If A, B independent, then P(A|B)=?
A
P(A) (B’s occurrence doesn’t affect A).
24
Q
Flip a fair coin 3 times: sample space size?
A
2³=8 outcomes.
25
Flip 3 fair coins: P(exactly 2 heads)?
Number ways=3 (HHT, HTH, THH). So 3/8.
26
Roll 2 fair dice: total outcomes =?
6×6=36.
27
Roll 2 fair dice: P(sum=7)?
Number of ways=6 (1+6,2+5,3+4,4+3,5+2,6+1), so 6/36=1/6.
28
Roll 2 fair dice: P(sum=12)?
Only (6,6) => 1/36.
29
Define random variable X in probability.
A numerical outcome assigned to each element in the sample space.
30
Bernoulli trial: P(success)=p => P(X=1)=? P(X=0)=?
P(X=1)=p, P(X=0)=1−p.
31
Binomial distribution: P(X=k) formula for n trials, success prob p?
P(X=k)= (nCk) p^k (1−p)^(n−k).
32
Mean of a Binomial(n,p) random variable?
μ= n p.
33
Variance of a Binomial(n,p) random variable?
σ²= n p (1−p).
34
Define uniform probability distribution on {1,2,...,n}.
Each outcome has probability 1/n.
35
If X~Binomial(4,0.5), P(X=2)=?
(4C2)(0.5)²(0.5)²= 6×(0.25)×(0.25)= 6×0.0625=0.375.
36
A fair coin (p=0.5) tossed 4 times => total heads. Probability of exactly 3 heads?
(4C3)(0.5)³(0.5)¹=4×0.125×0.5=0.25.
37
Poisson distribution formula for P(X=k), parameter λ>0?
P(X=k)= (λ^k e^(−λ))/ k!.
38
Mean of a Poisson(λ) random variable is...?
λ (also variance = λ).
39
Conditional probability formula again: P(A|B)=?
P(A and B)/P(B).
40
Define complement rule in probability: P(Aᶜ)=?
1 − P(A).
41
If event A is ‘die shows prime number’, prime faces are {2,3,5}, P(A)=?
3/6=1/2.
42
Odds in favor of event A is ratio P(A)/P(Aᶜ). If P(A)=1/3 => odds=?
1/3 : 2/3=1:2.
43
Law of Total Probability: P(B)= Σ P(B|Aᵢ)P(Aᵢ). T/F?
True, summing over partition events Aᵢ.
44
If P(A)=0.3, P(B)=0.3, P(A and B)=0.09 => Are A, B independent?
Check 0.3×0.3=0.09 => yes, they match => independent.
45
Bayes’ Theorem short form: P(A|B)=?
[P(B|A) P(A)] / P(B).
46
Probability that a random digit (0–9) is even?
5 even digits (0,2,4,6,8) => 5/10=1/2.
47
2 cards drawn from standard 52 deck (without replacement). P(both aces)?
(4/52)(3/51)= (1/13)(3/51)=3/663=1/221.
48
A single draw from deck: P(face card) if face=Jack,Queen,King => 12 face cards. Probability?
12/52=3/13.
49
If events are collectively exhaustive, their probabilities sum to...?
1.
50
If P(A and B)=0 => A, B are disjoint => P(A or B)=?
P(A)+P(B).
51
Dice: sum≥10 => sums 10,11,12. Number of combos? 10 => 3 combos, 11 =>2 combos, 12 =>1 combo => total 6. Probability=?
6/36=1/6.
52
Random coin flips 2 times: Probability 0 heads =>?
(1/2)×(1/2)=1/4 => TT outcome.
53
Random coin flips 2 times: Probability 2 heads =>?
1/4 => HH only.
54
Random coin flips 2 times: Probability at least 1 head =>?
3/4 => HH,HT,TH.
55
If P(E)=0.2 => odds against E is...?
0.8 : 0.2=4:1.
56
If P(A|B)=0.7 => interpret?
Given B occurred, chance of A is 0.7.
57
If X is a discrete random variable, sum of P(X=x) over all x=?
1.
58
Expected value E(X) formula for discrete X=?
Σ x·P(X=x).
59
Var(X)= E(X²) − [E(X)]² => T/F?
True, basic variance formula.
60
In a fair coin, P(H)=P(T)=0.5 => expected number of heads in 1 toss=?
0.5.
61
If we do 10 coin flips, number of heads ~ Binomial(10,0.5). Probability of 5 heads=?
(10C5)(0.5)^(10)= 252×(1/1024)= 252/1024 ≈ 0.246.
62
Geometric distribution’s first success at trial k => formula?
P(X=k)= (1−p)^(k−1) p (for k=1,2,...).
63
In rolling one fair die, P(prime face)? (2,3,5) => 3 outcomes => P=?
3/6=1/2.
64
Choose 2 from 4 objects. # ways=?
(4C2)=6 => if equally likely, each pair has prob 1/6.
65
2 children in a family. Probability both are girls (assuming boy/girl equally likely)?
1/4 => outcomes: (G,G),(G,B),(B,G),(B,B).
66
Deck of 52, probability of drawing spade?
13/52=1/4.
67
Deck of 52, probability card is spade or heart?
26/52=1/2 => half the deck.
68
If P(A)=0.7, P(B)=0.6, P(A and B)=0.3 => P(A or B)=?
0.7+0.6−0.3=1.0.
69
From 3 coin flips (8 outcomes), P(exactly 1 head) =>?
3 out of 8 =>3/8=0.375.
70
Spin a fair spinner with 4 equal sectors. Probability of landing on sector #3 =>?
1/4.
71
Toss 2 fair coins => sample space=4. Probability 2 heads =>?
1/4 => only (H,H).
72
Random integer from 1 to 100 => probability multiple of 10 =>?
10 multiples => 10/100=1/10.
73
Random integer from 1 to 100 => P(multiple of 3 or 5) using inclusion-exclusion => # of multiples 3=33, 5=20, 15=6 => total=?
33+20−6=47 => P=47/100.
74
An event has P=0.45 => interpret odds in favor => 0.45:0.55 => ~0.82:1 => T/F?
Yes, that’s correct ratio.
75
Roll one fair die => P(≥4) => outcomes {4,5,6} => 3 out of 6 =>?
1/2.
76
Probability that a random letter from English alphabet (26 letters) is vowel if vowels are {A,E,I,O,U}? => 5 out of 26 =>?
5/26.
77
2 events A,B independent => P(A and B)= 0.12 => P(A)=0.3 => P(B)=?
0.4 => because 0.3×B=0.12 => B=0.4.
78
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?
True => 5/16=0.3125.
79
In a group of 10, pick 3 for committee => # ways=(10C3)=120 => Probability of a specific group of 3 is 1/120 => T/F?
True.
80
If P(A|B)= 0.3 => interpret as: when B occurs, A occurs 30% of the time => T/F?
True.
81
For Poisson(λ=2), P(X=0)= e^(−2)(2^0)/0!= e^(−2) => T/F?
True => about 0.1353.
82
Mean of Binomial(10,0.3)= n p => 3 => T/F?
Yes => 3.
83
Variance of Binomial(10,0.3)= n p(1−p)= 10×0.3×0.7= 2.1 => T/F?
True.
84
Dice sum problem: P(sum=2) => only (1,1) => 1/36 => T/F?
Yes => correct.
85
Dice sum problem: P(sum=3) => (1,2),(2,1)=2/36 => T/F?
True => 1/18.
86
Card deck: Probability of red card => half the deck => 26/52=1/2 => T/F?
True.
87
Card deck: Probability of drawing a face card => J,Q,K => 4 each suit => 12 total => 12/52=3/13 => T/F?
True.
88
If (1,2),(2,1)=2/36 => T/F?
True => 1/18.
89
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?
True.
90
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?
True => they are independent.
91
Saying probability of an event is 0.2 => means 20% chance => T/F?
True.
92
Conditional formula again: P(A and B)= P(A|B)×P(B) => T/F?
True.
93
Pick one letter from ‘MATH’ => 4 letters => P(A)= Probability letter is ‘A’ => 1/4 => T/F?
True.
94
Probability of picking a random number from 1 to 10 that is multiple of 2 => 5 out of 10 => T/F?
True => 1/2.
95
In 3 coin flips, P(TTT)= (1/2)³=1/8 => T/F?
True.
96
In 3 coin flips, P(no heads)= 1/8 => T/F?
True => same scenario as TTT.
97
If two events are exhaustive => P(A or B)=1 => T/F?
True => they cover entire sample space.
98
If P(A)=0 => event A is impossible => T/F?
True.
99
Pick random digit 0–9 => Probability of <5 => digits {0,1,2,3,4} => 5 out of 10 => T/F?
True => 1/2.
100
Random day from week => Probability day is Monday => 1/7 => T/F?
True, equally likely assumption.
101
If we want P(A or B or C) for 3 events => we can use inclusion-exclusion => T/F?
True => P(A)+P(B)+P(C)−(pairs) + P(A and B and C).
102
What is the formula for the probability of the complement of event A?
P(Aᶜ)=1−P(A).
103
State the formula for the union of two events A and B (general addition rule).
P(A ∪ B)=P(A)+P(B)−P(A ∩ B).
104
Formula for A, B disjoint: P(A ∪ B)=?
P(A)+P(B), because P(A ∩ B)=0.
105
Define conditional probability P(A|B).
P(A|B)= P(A ∩ B)/P(B), if P(B)≠0.
106
When are events A and B independent in terms of probabilities?
P(A ∩ B)= P(A)·P(B).
107
Formula for conditional probability if A, B independent?
P(A|B)= P(A).
108
Law of Total Probability formula for P(B) if {Aᵢ} partition the sample space?
P(B)= Σ P(B|Aᵢ)P(Aᵢ).
109
Bayes’ Theorem formula in short?
P(A|B)= [P(B|A) P(A)] / P(B).
110
Odds in favor of A in terms of P(A)?
Odds(A)= P(A) : [1−P(A)].
111
Odds against A if P(A)=p =>?
(1−p) : p.
112
Number of ways to choose k items from n: formula for nCk?
(nCk)= n! / [k! (n−k)!].
113
Define the sample space size for n coin flips.
2ⁿ.
114
Probability of exactly k heads in n fair coin flips?
(nCk)(1/2)ⁿ.
115
Formula for the binomial distribution P(X=k) with parameters n, p?
P(X=k)= (nCk) p^k (1−p)^(n−k).
116
Mean of a binomial random variable X ~ Binomial(n,p)?
E[X]=n p.
117
Variance of a binomial X ~ Binomial(n,p)?
Var(X)= n p (1−p).
118
Geometric distribution: P(X=k) formula for first success on kth trial (p=success prob)?
P(X=k)= (1−p)^(k−1) p, (k=1,2,...).
119
Mean of geometric distribution with parameter p?
1/p.
120
Poisson distribution formula for P(X=k) with rate λ>0?
P(X=k)= (λ^k e^(−λ))/k!.
121
Mean (and also variance) of Poisson(λ) distribution?
λ.
122
Empirical probability definition formula (based on trials)?
P(A)= (Number of times A occurs)/(Total trials).
123
Complement rule formula again: P(Aᶜ)=?
1−P(A).
124
Conditional probability rearranged: P(A ∩ B)=?
P(A|B) P(B)= P(B|A) P(A).
125
If random variable X has outcomes xᵢ, with probabilities pᵢ, sum pᵢ=1 => mean formula?
E[X]= Σ xᵢ pᵢ.
126
Variance formula for discrete X: Var(X)=?
Σ (xᵢ² pᵢ) − [E(X)]².
127
If events A and B are independent => formula for P(A or B)?
P(A)+P(B)−P(A)P(B).
128
General multiplication rule: P(A ∩ B)=?
P(A) P(B|A)= P(B) P(A|B).
129
Uniform probability distribution on n equally likely outcomes => probability of each outcome=?
1/n.
130
Permutation formula for n distinct items taken k at a time?
P(n,k)= n! / (n−k)!.
131
Combination formula for n distinct items choose k => (nCk) again?
n! / [k!(n−k)!].
132
Hypergeometric distribution formula concept: Probability of k successes from K available in N, drawing n?
P(X=k)= [(KCk)((N−K)C(n−k))]/ (NCn).
133
Bernoulli random variable X with success prob p => P(X=1)=p => formula for mean, variance?
E[X]=p, Var(X)=p(1−p).
134
If events A₁,...,Aₙ partition sample space => P(B)=?
Σ P(B|Aᵢ) P(Aᵢ). (Law of total probability).
135
If X~Binomial(n,p), Probability P(X=0)=?
(nC0) p^0 (1−p)ⁿ= (1−p)ⁿ.
136
If X~Poisson(λ), Probability P(X=0)=?
e^(−λ).
137
In random experiment, sum of probabilities of all disjoint events covering S=?
1.
138
Continuous uniform distribution on [a,b], pdf f(x)=?
1/(b−a) for x in [a,b].
139
Mean of continuous uniform [a,b] is?
(a+b)/2.
140
If X~Exp(λ), pdf is λ e^(−λx). Probability formula for X≥0 => T/F?
True => That’s exponential distribution.
141
Exponential(λ) distribution mean=?
1/λ.
142
Conditional probability formula for P(B|A)= P(B ∩ A)/P(A). T/F?
True.
143
Bayes formula: P(A|B)= [P(B|A)P(A)] / P(B). T/F?
True.
144
When do we say A and B are mutually exclusive?
If P(A ∩ B)=0.
145
When do we say A and B are collectively exhaustive?
If A ∪ B covers entire sample space => P(A∪B)=1.
146
If sample space S has N equally likely outcomes => P(A)= #A / N => T/F?
True.
147
Define joint probability of events (A,B).
P(A∩B)= Probability that both events occur.
148
Define marginal probability from joint distribution: P(A)= Σ or ∫ over B => T/F?
True.
149
Conditional pmf for discrete: p(x|y)= p(x,y)/ p(y)? T/F?
True.
150
In random selection of 2 from n, # ways is (nC2). Probability formula can use that T/F?
True.
151
Odds in favor of event E is P(E)/P(Eᶜ). T/F?
True.
152
Flip a fair coin twice. What’s the probability of 2 heads?
Outcomes: HH, HT, TH, TT ⇒ 4 total ⇒ only HH is success ⇒ 1/4.
153
Flip a fair coin twice. Probability of at least 1 head?
3 out of 4 ⇒ 3/4.
154
Roll one fair six-sided die. Probability of rolling a 5 or 6?
2/6=1/3.
155
Roll two fair dice. Probability that sum=2?
Only (1,1) ⇒ 1/36.
156
Roll two fair dice. Probability that sum=7?
6 outcomes => 6/36=1/6.
157
Random integer from 1 to 10. Probability it’s even?
Even: {2,4,6,8,10} => 5 out of 10 => 1/2.
158
Random integer from 1 to 10. Probability it’s prime?
Primes= {2,3,5,7} => 4 out of 10 => 2/5.
159
Deck of 52 cards. Probability of drawing a heart?
13 hearts => 13/52=1/4.
160
Deck of 52 cards. Probability card is a heart or spade?
26/52=1/2.
161
Deck of 52, Probability card is an Ace?
4 out of 52 => 1/13.
162
Drawing one card: Probability card is red face card?
6 out of 52 => 6/52=3/26.
163
2 cards drawn from 52 without replacement. Probability both are Aces?
(4/52)(3/51)= (1/13)(3/51)= 3/663=1/221.
164
2 fair coins. Probability exactly 1 head?
HT or TH => 2 out of 4 => 1/2.
165
Pick random day from 7-day week. Probability of it being Monday?
1/7.
166
Choose 2 from 5 distinct items at random. Probability the pair includes a specific item X?
Ways total= (5C2)=10. Pairs including X => choose 1 from remaining 4 => (4C1)=4 => Probability=4/10=2/5.
167
Spin a fair spinner with 5 equal sectors. Probability it lands on sector #3?
1/5.
168
Two fair dice rolled. Probability that sum≥10?
6 outcomes => 6/36=1/6.
169
Pick a random digit 0–9. Probability digit is at least 5?
True.
170
Random integer 1–20. Probability multiple of 4?
Yes => 1/4.
171
Roll one fair die. Probability prime face?
Correct.
172
One fair coin, toss 1 time. Probability heads=?
1/2.
173
Random letter from ‘CAT’. Probability letter is ‘A’?
True.
174
Pick 1 from {A,B,C,D} equally likely. Probability of picking C?
1/4.
175
2 coin flips: Probability 2 tails?
Yes.
176
Random integer from 1–6. Probability ≤2?
Yes.
177
Deck of 52, Probability that drawn card is black King?
Correct.
178
2 draws from deck without replacement. Probability 2 spades?
True.
179
Probability that random chord of a circle is > radius leads to paradoxical results. This is known as the Bertrand paradox?
True.
180
Pick random day from {Mon,Tue,Wed,Thu,Fri,Sat,Sun}. Probability weekend?
True.
181
A single coin toss: Probability not heads?
True.
182
Spin fair spinner with 8 equal regions. Probability region #7?
True.
183
Roll fair six-sided die. Probability (even number)?
True.
184
Binomial example: 3 coin flips. Probability exactly 2 heads?
True.
185
Pick random integer from 1–4. Probability prime?
True.
186
2 coin flips. Probability at least 1 tail?
Yes.
187
Flip fair coin 3 times. Probability all heads?
True.
188
Draw 1 from {apple, banana, cherry}, each equally likely. Probability banana?
True.
189
Deck 52. Probability is a heart or diamond?
Correct.
190
Random digit 0–9. Probability digit ≥8?
True.
191
Random letter from ‘HELLO’: Probability letter=‘L’?
True.
192
Dice sum problem: Probability sum=6?
Yes.
193
Dice sum=8?
True.
194
Probability of 3 heads in 4 tosses?
True.
195
Pick random integer from 1–12. Probability multiple of 3?
True.
196
Random choice from {red,blue,green}. Probability is green?
True.
197
2 draws from 52 deck w/o replacement. Probability 1st is spade, 2nd is heart?
Yes.
198
One fair coin. P(Heads)?
1/2.
199
1 draw from deck. Probability a black ace?
True.
200
One 6-sided die. Probability face≥5?
True.
201
Pick 1 from {0,1}. Probability pick 1?
True.
202
Flip a fair coin 3 times. Probability exactly 1 head?
Number ways = 3 (HTT, THT, TTH), total outcomes=8 ⇒ 3/8.
203
Flip a fair coin 3 times. Probability at least 1 tail?
Complement is no tails => all heads => 1/8 ⇒ so 1−1/8=7/8.
204
Roll one fair six-sided die. Probability of rolling a 1 or 2?
2 out of 6 ⇒ 1/3.
205
Roll two fair dice. Probability that sum=4?
Sum=4 combos: (1,3),(2,2),(3,1) ⇒ 3 combos ⇒ 3/36=1/12.
206
Random integer from 1 to 5. Probability equals 3 => ?
1 out of 5 ⇒ 1/5.
207
Pick a random digit 0–9. Probability even?
Even digits = {0,2,4,6,8} => 5 out of 10 ⇒ 1/2.
208
Pick a random digit 0–9. Probability prime (2,3,5,7)?
4 out of 10 ⇒ 2/5.
209
Deck of 52. Probability spade or heart?
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
Deck of 52. Probability face card? (J,Q,K => 4 each suit =>12 total)
12/52=3/13.
211
Random letter from the word ‘GEEK’ => 4 letters (G,E,E,K). Probability letter= E =>?
2 out of 4 => 1/2.
212
Toss 2 fair coins: Probability 2 tails =>?
(1/2)×(1/2)=1/4 => TT outcome.
213
If an event has P(A)=0.7 => P(not A)=?
0.3 => complement rule.
214
Two events A,B disjoint => P(A or B)= P(A)+P(B). T/F?
True => no overlap.
215
Sum of probabilities of all possible outcomes in a discrete sample space => ?
1.
216
Probability an odd face from one fair die => {1,3,5} => 3/6=1/2 => T/F?
True.
217
Two fair dice, Probability sum=3 => combos (1,2),(2,1) => 2/36=1/18 => T/F?
True.
218
Select random day of 7. Probability it’s Wed =>?
1/7.
219
Random letter from ‘MISS’: 4 letters => M,I,S,S => Probability letter= S => 2/4=1/2 => T/F?
True.
220
Pick integer from 1 to 6. Probability prime => {2,3,5} => 3/6=1/2 => T/F?
True.
221
Flip fair coin 4 times => total outcomes=2⁴=16 => Probability all tails => 1/16 => T/F?
True.
222
3 coin flips: Probability exactly 2 tails => (3C2)(1/2)³= 3×(1/8)=3/8 => T/F?
True.
223
Deck of 52: Probability of a Joker =>?
Typically 0, since standard deck has no Joker (or assume 52 excludes them).
224
Random integer from 1 to 100 => Probability multiple of 10 => multiples are 10,20,...,100 => 10 out of 100 =>1/10 => T/F?
True.
225
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?
True.
226
Pick random letter from ‘ABC’. Probability letter= ‘C’? =>1/3 => T/F?
True.
227
Random coin toss: Probability heads =>1/2 => T/F?
True => fair coin.
228
2 draws from deck without replacement => Probability 2 hearts => (13/52)(12/51)= 13×12/(52×51)=> T/F?
Yes => that's correct.
229
Sample space with n equally likely outcomes => Probability of an event A is #(A)/n => T/F?
True.
230
When flipping 1 fair coin, P(H)=0.7 => is that correct?
No, for a fair coin P(H)=0.5 => 0.7 would be unfair.
231
Pick integer 1–8 => Probability is >5 => outcomes {6,7,8} =>3/8 => T/F?
True => 3 out of 8.
232
Dice problem: Probability sum=8 => combos (2,6),(3,5),(4,4),(5,3),(6,2)=5 => 5/36 => T/F?
True.
233
If P(A)=0.3, P(B)=0.5, A,B independent => P(A∩B)= (0.3)(0.5)=0.15 => T/F?
True => that's the definition.
234
Probability of picking red card from deck => 26 red cards => 26/52=1/2 => T/F?
True.
235
Probability of picking the Queen of hearts => 1/52 => T/F?
True.
236
Random choice from {red,green,blue} => Probability ‘blue’ =>1/3 => T/F?
Yes => 1/3.
237
If event E has P(E)=0 => E is impossible => T/F?
True.
238
Two disjoint events A,B => P(A∩B)=0 => T/F?
True.
239
In a single die roll, Probability #≥5 => {5,6}=2 out of 6 =>1/3 => T/F?
True.
240
Pick integer from 1–12 => Probability multiple of 4 => {4,8,12}=3 => 3/12=1/4 => T/F?
Yes => 1/4.
241
Probability ‘2 heads in 2 coin flips’ =>1/4 => T/F?
True => HH only.
242
Probability exactly 3 heads in 3 coin flips =>(3C3)(1/2)³=1×1/8=1/8 => T/F?
True => HHH.
243
Pick from 1–4 => Probability prime => {2,3}=2/4=1/2 => T/F?
Yes.
244
Deck of 52 => Probability card is King => 4/52=1/13 => T/F?
True.
245
In drawing 1 from {a,b,c,d}, Probability (not c)=> 3/4 => T/F?
True => complement approach.
246
Roll 2 dice => Probability sum=9 => combos= (3,6),(4,5),(5,4),(6,3)=4 => 4/36=1/9 => T/F?
True.
247
If P(A)=0.6 => P(Aᶜ)= 0.4 => T/F?
True => complement rule.
248
Probability event A occurs at least once in 2 trials =>1− P(A never occurs) => T/F?
Yes => correct approach.
249
A fair coin tossed once: P(Tails)= 1/2 => T/F?
True.
250
Pick random integer from 1–100 => Probability it’s 1 => 1/100 => T/F?
Yes => 1 out of 100.
251
Pick random letter from ‘MATH’ => Probability letter ‘T’? => {M,A,T,H} => T =>1/4 => T/F?
True => equally likely.
