A fair coin is flipped once and a fair six-sided die is rolled once. How many outcomes are in the sample space for this combined experiment?
12
✓
8
36
6
Correct Answer
12
The coin has 2 outcomes and the die has 6, and each coin result can pair with each die result, so the sample space has 2 x 6 = 12 outcomes, such as H1 or T4. The value 8 comes from adding 2 and 6 instead of multiplying. The value 36 is the sample space for rolling two dice. The value 6 counts only the die.
Category: Basic Terminology
Question 2
A fair six-sided die is rolled once. What is P(the roll is even OR the roll is greater than 4)?
5/6
2/3
✓
1/3
1/2
Correct Answer
2/3
The even rolls are 2, 4 and 6, and the rolls greater than 4 are 5 and 6, so the union is {2, 4, 5, 6}, which is 4 of 6 equally likely outcomes, or 2/3. The value 5/6 adds 3/6 and 2/6 without subtracting the roll of 6 that was counted twice. The value 1/3 is only the probability of rolling greater than 4. The value 1/2 is only the probability of an even roll.
Category: Addition Rule
Question 3
For an event A, P(A) = 0.35. What is P(A'), the probability of the complement of A?
0.35
0.70
0.65
✓
1.35
Correct Answer
0.65
An event and its complement together cover the whole sample space, so the probability of A' is 1 - 0.35 = 0.65. The value 0.35 repeats the probability of A itself. The value 0.70 doubles the probability of A, which has no meaning here. The value 1.35 adds 1 instead of subtracting and is above 1, so it cannot be a probability.
Category: Basic Terminology
Question 4
Events A and B have P(A) = 0.4, P(B) = 0.5 and P(A AND B) = 0.2. What is P(A OR B)?
0.9
0.2
0.3
0.7
✓
Correct Answer
0.7
The addition rule adds the two event probabilities and subtracts the overlap: 0.4 + 0.5 - 0.2 = 0.7. Subtracting the overlap keeps outcomes in both events from being counted twice. The value 0.9 forgets to subtract the overlap. The value 0.2 is P(A AND B), not P(A OR B). The value 0.3 is the probability that neither event occurs.
Category: Addition Rule
Question 5
Event A has probability 0.32 and event B has probability 0.46, and A and B are mutually exclusive. What is the probability that A or B occurs?
0.78
✓
0.1472
0.14
0.22
Correct Answer
0.78
Mutually exclusive events have no outcomes in common, so P(A AND B) = 0 and P(A OR B) = 0.32 + 0.46 = 0.78. The value 0.1472 multiplies the probabilities, which would give P(A AND B) only for independent events. The value 0.14 is the difference of the two probabilities. The value 0.22 is the probability that neither event occurs.
Category: Addition Rule
Question 6
Events A and B are independent, with P(A) = 0.5 and P(B) = 0.6. What is P(A AND B)?
1.1
0.3
✓
0.83
0.1
Correct Answer
0.3
For independent events, the probability that both occur is the product of their probabilities: 0.5 x 0.6 = 0.3. The value 1.1 adds the probabilities and exceeds 1, so it cannot be a probability. The value 0.83 divides 0.5 by 0.6, which is not a rule for joint probability. The value 0.1 is the difference of the two probabilities.
Category: Multiplication Rule
Question 7
For events A and B, P(A AND B) = 0.15 and P(B) = 0.6. What is P(A|B)?
0.09
0.45
0.25
✓
0.75
Correct Answer
0.25
Conditional probability divides the joint probability by the probability of the given event: P(A|B) = 0.15 / 0.6 = 0.25. The value 0.09 multiplies 0.15 by 0.6 instead of dividing. The value 0.45 subtracts 0.15 from 0.6. The value 0.75 is 1 - 0.25, the probability that A does not occur given B.
Category: Conditional Probability
Question 8
Which statement describes two mutually exclusive events?
One does not change the chance of the other
Their probabilities must add to 1
Each must have a probability of 0.5
They cannot occur on the same trial
✓
Correct Answer
They cannot occur on the same trial
Mutually exclusive events share no outcomes, so they cannot happen at the same time and P(A AND B) = 0. The statement that one does not change the chance of the other describes independence, a different idea. Probabilities that must add to 1 describe an event and its complement. Equal probabilities of 0.5 are not required for events to be mutually exclusive.
Category: Independence and Exclusivity
Question 9
Events A and B have P(A) = 0.3, P(B) = 0.6 and P(A AND B) = 0.18. How are A and B related?
Independent but not mutually exclusive
✓
Mutually exclusive but not independent
Both independent and mutually exclusive
Neither independent nor mutually exclusive
Correct Answer
Independent but not mutually exclusive
Because the product of the two event probabilities, 0.3 x 0.6 = 0.18, equals P(A AND B), the events are independent, and because P(A AND B) is greater than 0, they share outcomes and are not mutually exclusive. Mutually exclusive but not independent would need P(A AND B) = 0. Both independent and mutually exclusive is not possible here, since the overlap is not 0. Neither independent nor mutually exclusive would need the product to differ from 0.18.
Category: Independence and Exclusivity
Question 10
A bag holds 5 red marbles and 3 blue marbles. Two marbles are drawn one after the other without replacement. What is P(both marbles are red)?
25/64
5/14
✓
5/8
1/4
Correct Answer
5/14
The first draw is red with probability 5/8, and after one red is removed, 4 of the remaining 7 marbles are red, so P(both red) = 5/8 x 4/7 = 20/56 = 5/14. The value 25/64 is the answer when the first marble is replaced, which keeps the second draw at 5/8. The value 5/8 is only the chance that the first marble is red. The value 1/4 does not come from the counts in the bag.
Category: Multiplication Rule
Question 11
A card is drawn from a standard deck, put back, and the deck is shuffled before a second card is drawn. How are the results of the two draws related?
They are dependent
They are mutually exclusive
They are independent
✓
They are complements
Correct Answer
They are independent
Replacing the card restores the deck to its original state, so the result of the first draw does not change any probability for the second draw. Dependent draws occur when sampling is done without replacement. Mutually exclusive would mean the two results could not both occur, but both draws can be hearts. Complements would mean exactly one of the two events must occur, which does not describe two separate draws.
Category: Independence and Exclusivity
Question 12
A survey of 200 adults records age group and whether each person exercises regularly. Under 40: 60 exercise and 40 do not. Age 40 and over: 50 exercise and 50 do not. One adult is chosen at random. What is P(under 40 | exercises regularly), rounded to two decimal places?
0.60
0.30
0.50
0.55
✓
Correct Answer
0.55
Given that the person exercises, the sample space shrinks to the 60 + 50 = 110 people who exercise, and 60 of them are under 40, so the probability is 60/110, about 0.545, which rounds to 0.55. The value 0.60 reverses the condition, giving P(exercises | under 40) = 60/100. The value 0.30 is the joint probability 60/200. The value 0.50 is the share of all 200 adults who are under 40.
Category: Contingency Tables
Question 13
A class of 100 students is classified by housing and pet ownership. Of the 40 who live on campus, 10 own a pet. Of the 60 who live off campus, 35 own a pet. If one student is picked at random, what is P(lives on campus OR owns a pet)?
0.75
✓
0.85
0.10
0.25
Correct Answer
0.75
There are 40 on-campus students and 10 + 35 = 45 pet owners, and 10 students are in both groups, so P(on campus OR pet) = (40 + 45 - 10) / 100 = 75/100 = 0.75. The value 0.85 adds 40 and 45 without removing the 10 students counted twice. The value 0.10 is the probability of being in both groups. The value 0.25 is the probability of being in neither group, the 25 off-campus students with no pet.
Category: Contingency Tables
Question 14
On any day, P(rain) = 0.3. If it rains, a student is late with probability 0.5; if it does not rain, the student is late with probability 0.1. What is the probability the student is late on a randomly chosen day?
0.60
0.22
✓
0.15
0.07
Correct Answer
0.22
A tree diagram has two paths that end in being late: rain then late, 0.3 x 0.5 = 0.15, and no rain then late, 0.7 x 0.1 = 0.07. Adding the paths gives 0.15 + 0.07 = 0.22. The value 0.60 adds the two conditional probabilities, which is not valid. The value 0.15 counts only the rainy path. The value 0.07 counts only the dry path.
Category: Tree and Venn Diagrams
Question 15
On any day, P(rain) = 0.3. If it rains, a student is late with probability 0.5; if it does not rain, the student is late with probability 0.1. Given that the student was late, what is the probability that it rained, rounded to two decimal places?
0.50
0.30
0.68
✓
0.15
Correct Answer
0.68
The overall chance of being late is 0.3 x 0.5 + 0.7 x 0.1 = 0.22, and the rain-and-late path is 0.15, so P(rain | late) = 0.15 / 0.22, about 0.682, which rounds to 0.68. The value 0.50 is P(late | rain), the condition reversed. The value 0.30 is the chance of rain before knowing the student was late. The value 0.15 is the joint probability of rain and late, not a conditional probability.
Category: Conditional Probability
Question 16
A fair coin is flipped thousands of times. What does the law of large numbers predict?
The count of heads will exactly equal the count of tails
Heads becomes more likely after a run of tails
The proportion of heads moves farther from 0.5
The proportion of heads gets closer to 0.5
✓
Correct Answer
The proportion of heads gets closer to 0.5
The law of large numbers says that as an experiment is repeated more and more, the relative frequency of an outcome tends to approach its theoretical probability, which is 0.5 for heads. It does not promise that the count of heads will exactly equal the count of tails; the two counts can differ even when the proportion is near 0.5. The claim that heads becomes more likely after a run of tails is the gambler's fallacy, because fair flips are independent. The claim that the proportion moves farther from 0.5 is the opposite of what the law states.
Category: Basic Terminology
Question 17
Which experiment has outcomes that are equally likely?
Rolling a fair six-sided die
✓
Recording the sum of two fair dice
Flipping a coin weighted toward heads
Noting whether tomorrow is rainy or dry
Correct Answer
Rolling a fair six-sided die
A fair die gives each face, 1 through 6, the same probability of 1/6, so its outcomes are equally likely. The sum of two fair dice is not equally likely, since a sum of 7 can happen six ways and a sum of 2 only one way. A weighted coin favors one side by design. Rainy and dry weather days do not occur with equal probability in general.
Category: Basic Terminology
Question 18
What does P(A|B) represent?
The probability that A and B both occur
The probability of A, given that B has occurred
✓
The probability of B, given that A has occurred
The probability that A or B occurs
Correct Answer
The probability of A, given that B has occurred
The vertical bar is read as given, so P(A|B) is the probability of A when B is known to have happened, and the sample space is reduced to the outcomes in B. The probability that A and B both occur is the joint probability P(A AND B). The probability of B, given that A has occurred, is P(B|A), which usually differs from P(A|B). The probability that A or B occurs is P(A OR B), a union.
Category: Conditional Probability
Question 19
A fair coin is flipped 3 times. What is the probability of getting at least one head?
1/8
3/8
7/8
✓
1/2
Correct Answer
7/8
The only way to get no heads is TTT, which has probability (1/2) x (1/2) x (1/2) = 1/8, so by the complement rule P(at least one head) = 1 - 1/8 = 7/8. The value 1/8 is the probability of no heads, the complement itself. The value 3/8 is the probability of exactly one head. The value 1/2 is the probability of a head on a single flip.
Category: Multiplication Rule
Question 20
In a Venn diagram with one circle for event A and one for event B, which region shows the event A AND B?
Every point inside either circle
The area outside both circles
The part of circle A outside circle B
The overlap of the two circles
✓
Correct Answer
The overlap of the two circles
A AND B contains only outcomes that belong to both events, which is where the two circles overlap. Every point inside either circle shows A OR B, the union. The area outside both circles shows outcomes in neither event. The part of circle A outside circle B shows outcomes in A but not in B.
Category: Tree and Venn Diagrams