The sample space of an experiment is the set of all of its possible outcomes.
Correct Answer
True
True. The sample space, often written S, lists every outcome that can occur, such as {H, T} for one coin flip. Every event is built from outcomes in the sample space.
Category: Basic Terminology
Question 2
A probability can be any number from -1 to 1.
Correct Answer
False
False. A probability is always between 0 and 1, inclusive. A probability of 0 means the event cannot occur, and a probability of 1 means it is certain; negative values have no meaning.
Category: Basic Terminology
Question 3
If events A and B are independent and P(B) is greater than 0, then P(A|B) equals P(A).
Correct Answer
True
True. Independence means that knowing B occurred does not change the probability of A, so P(A|B) is the same as the unconditional probability of A. This is one of the equivalent ways to check whether two events are independent.
Category: Independence and Exclusivity
Question 4
When two fair six-sided dice are rolled, the probability that the sum is 7 is 1/36.
Correct Answer
False
False. Two dice have 36 equally likely outcomes, and six of them give a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). The probability is 6/36 = 1/6; 1/36 is the probability of one specific pair, such as double sixes.
Category: Basic Terminology
Question 5
Rolling a fair die and flipping a fair coin are independent events.
Correct Answer
True
True. The die and the coin are separate physical processes, so the result of one cannot change the probabilities for the other. For example, P(heads AND a 6) = (1/2)(1/6) = 1/12.
Category: Independence and Exclusivity
Question 6
For any two events A and B, P(A|B) equals P(B|A).
Correct Answer
False
False. P(A|B) divides P(A AND B) by the probability of B, while P(B|A) divides it by the probability of A, so the two are equal only when A and B have the same probability (or never occur together). Confusing the two is a common error, such as mixing up the chance of a positive test given disease with the chance of disease given a positive test.
Category: Conditional Probability
Question 7
In a tree diagram, the probabilities on all branches leaving the same point add to 1.
Correct Answer
True
True. The branches leaving one point list every possible result at that stage, and those results are mutually exclusive, so their probabilities sum to 1. This is a quick way to check that a tree is set up correctly.
Category: Tree and Venn Diagrams
Question 8
The law of large numbers says that a short series of trials will closely match the theoretical probability.
Correct Answer
False
False. The law of large numbers is about the long run: as the number of repetitions grows, the relative frequency tends to get closer to the theoretical probability. A short series can differ a great deal from the theoretical value.
Category: Basic Terminology
Question 9
When all outcomes are equally likely, the probability of an event equals the number of outcomes in the event divided by the total number of outcomes.
Correct Answer
True
True. With equally likely outcomes, each one carries the same share of probability, so an event's probability is its count of outcomes over the size of the sample space. For example, rolling an even number on a fair die has probability 3/6 = 1/2.
Category: Basic Terminology
Question 10
Two mutually exclusive events that each have a positive probability must be independent.
Correct Answer
False
False. Such events are always dependent. Knowing one occurred makes the other impossible, so P(A AND B) = 0, while multiplying their two positive probabilities gives a number greater than 0, and the two do not match.
Category: Independence and Exclusivity
Question 11
In a contingency table, the row totals and the column totals each add up to the grand total.
Correct Answer
True
True. Every observation falls in exactly one row and exactly one column, so summing the row totals or summing the column totals counts every observation once. Both sums equal the grand total.
Category: Contingency Tables
Question 12
Finding P(A|B) restricts attention to only the outcomes in event B.
Correct Answer
True
True. Conditioning on B reduces the sample space to the outcomes in B, and P(A|B) is the share of that reduced space that is also in A. That is why the probability of B is the divisor when P(A|B) is computed from P(A AND B).
Category: Conditional Probability
Question 13
In a sample space of 8 equally likely outcomes, each outcome has probability 1/4.
Correct Answer
False
False. With 8 equally likely outcomes, each one has probability 1/8. In general, when n outcomes are equally likely, each has probability 1/n.
Category: Basic Terminology
Question 14
After a fair coin lands tails five times in a row, heads is more likely than tails on the next flip.
Correct Answer
False
False. Flips of a fair coin are independent, so the next flip still has probability 0.5 for heads and 0.5 for tails. Believing a result is due after a streak is called the gambler's fallacy.
Category: Independence and Exclusivity
Question 15
An event and its complement can both occur on the same trial.
Correct Answer
False
False. The complement A' contains exactly the outcomes not in A, so an event and its complement are mutually exclusive and cannot occur together. Exactly one of them happens on every trial, which is why the probability of A and the probability of A' add to 1.
Category: Basic Terminology
Question 16
In a two-way table, P(A|B) is found by dividing the count in the cell for A AND B by the grand total.
Correct Answer
False
False. Dividing that cell count by the grand total gives the joint probability P(A AND B). To find P(A|B), divide the cell count by the total for B, because the condition limits attention to the observations in B.
Category: Contingency Tables
Question 17
In a tree diagram, the probability of reaching the end of a path is found by adding the probabilities along its branches.
Correct Answer
False
False. The probabilities along a path are multiplied, which applies the multiplication rule: the probability of the first branch times the conditional probability of the second. Adding is used across different paths that lead to the same result.
Category: Tree and Venn Diagrams
Question 18
The rule P(A AND B) = P(A|B) x P(B) holds whether or not A and B are independent.
Correct Answer
True
True. This general multiplication rule comes directly from the definition of conditional probability, so it works for any events where B has a probability greater than 0. For independent events, P(A|B) is just the probability of A, so the rule simplifies to multiplying the two event probabilities.
Category: Multiplication Rule
Question 19
In a Venn diagram, two mutually exclusive events are drawn as circles that do not overlap.
Correct Answer
True
True. Mutually exclusive events share no outcomes, so there is no region that belongs to both circles. Overlapping circles would show outcomes in A AND B.
Category: Tree and Venn Diagrams
Question 20
The probability of an event that is certain to happen is 1.
Correct Answer
True
True. An event that contains every outcome in the sample space is certain, and its probability is 1, the largest value a probability can take. For example, rolling a number from 1 to 6 on a standard die is certain.
Category: Basic Terminology