Quiz: Fundamentals of Probability and Independence — 14 domande

1. What is the main purpose of probability in studying a random experiment?

A. To eliminate all uncertainty from the experiment

B. To list every outcome in alphabetical order

C. To quantify how likely the possible outcomes are

D. To replace counting with exact prediction

Answer: (C)

Spiegazione

Probability is used to quantify the likelihood of outcomes produced by a random experiment. It does not remove randomness or make the outcome exactly predictable.

2. What does a conditional probability calculation do to the reference universe?

A. It ignores the condition and uses the original sample space

B. It restricts the universe to the cases where the condition is true

C. It replaces probabilities with frequencies in the full population

D. It expands the universe to include every possible experiment

Answer: (B)

Spiegazione

Conditional probability is computed after adding extra information, so the reference universe is restricted to the subset where the condition holds. The full sample space is not used unchanged.

3. How is a marginal frequency in a contingency table computed?

A. By dividing a cell count by the column total

B. By dividing a cell count by the row total

C. By adding the row total and the column total

D. By dividing the row or column total by the overall total

Answer: (D)

Spiegazione

A marginal frequency is the proportion of the whole population having a given value, so its denominator is the overall total. Dividing by a row or column total gives a conditional frequency instead.

4. In a contingency table, what does the conditional frequency f_{a1}(b1) represent?

A. The proportion of a1 among all individuals in the population

B. The proportion of b1 among individuals already having a1

C. The proportion of a1 and b1 among all cell entries

D. The total number of individuals in row a1

Answer: (B)

Spiegazione

Conditional frequency measures how common b1 is inside the subpopulation defined by a1. Its denominator is the marginal count for a1, not the total population.

5. What is a random experiment in probability?

A. A set containing only one possible result

B. A calculation that always produces the same result

C. A table that organizes counts into rows and columns

D. A procedure whose outcome cannot be predicted in advance

Answer: (D)

Spiegazione

A random experiment is a procedure with an outcome that cannot be predicted beforehand. A single possible result describes an elementary event, not the experiment itself.

6. What is an elementary event?

A. The procedure used to generate outcomes

B. The complete set of all outcomes

C. Any event with several possible outcomes

D. An event containing exactly one possible outcome

Answer: (D)

Spiegazione

An elementary event contains exactly one outcome from the universe. The complete set of all outcomes is the universe Ω, not an elementary event.

7. How is the conditional probability of A given B defined when P(B) is not zero?

A. P(A ∪ B) divided by P(B)

B. P(A ∩ B) divided by P(B)

C. P(B) divided by P(A ∩ B)

D. P(A) divided by P(B)

Answer: (B)

Spiegazione

Conditional probability is defined by P_B(A)=P(A∩B)/P(B) when P(B)≠0. This reflects restricting attention to the cases where B occurs.

8. What does P_B(A) represent in practical terms?

A. The probability of B after removing all outcomes in A

B. The probability of A and B being mutually exclusive

C. The probability of A inside the universe where B is known to occur

D. The probability of A in the original universe without any restriction

Answer: (C)

Spiegazione

P_B(A) means the probability of A given that B has occurred, so the universe is restricted to B. It is not the unrestricted probability of A.

9. What is the probability of a path in a weighted probability tree?

A. The difference between the two branch probabilities on the path

B. The product of the branch probabilities along that path

C. The sum of the branch probabilities at the first node only

D. The probability of the last branch only

Answer: (B)

Spiegazione

In a weighted tree, the probability of a path is obtained by multiplying the branch probabilities along that path. Adding branch probabilities is used only for branches leaving the same node.

10. What must be true for the probabilities of all branches leaving the same node in a weighted tree?

A. They must add up to 1

B. They must multiply to 1

C. They must all be equal

D. They must be greater than 0.5

Answer: (A)

Spiegazione

Branches leaving the same node represent all possible next choices from that condition, so their probabilities sum to 1. Equal branch probabilities are not required.

11. Which formula expresses the total probability of A using a partition by B and its complement?

A. P(A)=P(B)+P(B̄)

B. P(A)=P(A∪B)+P(A∪B̄)

C. P(A)=P(A∩B)×P(A∩B̄)

D. P(A)=P(A∩B)+P(A∩B̄)

Answer: (D)

Spiegazione

The total probability formula writes A as the union of two disjoint cases: through B and through B̄. Since the cases do not overlap, their probabilities add.

12. Why is tree inversion useful in probability trees?

A. It turns every event into an independent event

B. It helps recover a missing probability using the total probability relation

C. It changes the outcome space into a contingency table

D. It removes the need for conditional probabilities

Answer: (B)

Spiegazione

Tree inversion is used when a needed probability is not directly shown in the original tree, and the total probability relation helps recover it. It does not create independence or eliminate conditioning.

13. When are two events considered independent?

A. When they cannot occur together

B. When one event is a subset of the other

C. When knowing one does not change the probability of the other

D. When their union has probability 1

Answer: (C)

Spiegazione

Independence means that learning one event occurred does not change the probability of the other. If the events cannot occur together, that is mutual exclusivity, not independence.

14. What is the probability of the intersection of two independent events A and B?

A. P(A) × P(B)

B. P(A) + P(B)

C. P(A) - P(B)

D. P(A) / P(B)

Answer: (A)

Spiegazione

For independent events, the product rule gives P(A∩B)=P(A)×P(B). This is equivalent to saying the conditional probability equals the unconditional probability.