Question 1

Which of the following best describes the power set $$2^{\Omega} of a $$sample space $$\Omega$$?

Accepted Answer

The set of all subsets of $$\Omega$$, including the empty set and $$\Omega$$ itself.

Question 2

According to Kolmogorov's axioms, which property must the probability of any event $$A$$ satisfy?

Accepted Answer

$$P[A] \geq 0$$

Question 3

A random experiment consists of rolling a fair 6-sided die. What is the probability of getting an even number?

Accepted Answer

$$\frac{1}{2}$$

Question 4

Which axiom is required for probability spaces with infinite sample spaces but not for finite sample spaces?

Accepted Answer

$$\sigma-$$additivity

Question 5

If $$A$$ and $$B$$ are disjoint events in a probability space, what is $$P[A \cup B]$$?

Accepted Answer

$$P[A] + P[B]$$

Question 6

In the context of probability theory, what does it mean if $$P[A] = 0$$ but $$A 
eq \emptyset$$?

Accepted Answer

Event $$A is $$very improbable but not impossible.

Question 7

Which of the following is NOT a requirement for a set system $$\mathcal{F} to be a $$field of sets?

Accepted Answer

Closure under multiplication

Question 8

Akaike’s Information Criterion (AIC) is given by which formula?

Accepted Answer

$$AIC = -2l + 2K$$

Question 9

In a continuous uniform probability law on the interval $$0 \leq X \leq 1$$, what is $$P(X \u003e 0.6)$$?

Accepted Answer

$$0.4$$

Question 10

Why is the Borel $$\sigma-$$field used for continuous sample spaces instead of the power set $$2^{\Omega}$$?

Accepted Answer

Because $$2^{\Omega}$$ does not satisfy all probability axioms for continuous $$\Omega.$$

Question 11

Which principle suggests that models should be as simple as possible, but not simpler?

Accepted Answer

Ockham’s Razor

Question 12

Given a random experiment: rolling a 4-faced die twice, what is the cardinality of the sample space?

Accepted Answer

$$16$$

Question 13

Which of the following is true for a probability mass function (PMF) in a discrete sample space?

Accepted Answer

The PMF assigns a probability to each possible outcome.

Question 14

If $$A_1$$, $$A_2$$, \ldots$$ are disjoint events and A$$ = \bigcup_{n=1}^{\infty} A_n$$, what does the $$\sigma$$-additivity axiom state?

Accepted Answer

$$P[A] = \sum_{n=1}^{\infty} P[A_n]$$

Question 15

Which of the following is NOT a property of outcomes in a sample space?

Accepted Answer

Always equally probable

Question 16

In the context of constructing a probability space, what is the first step?

Accepted Answer

Understand the random experiment and define the set of all outcomes $$\Omega.$$

Question 17

What is the probability of selecting a point at random from the unit square $$[0,1] 	imes [0,1]$$ such that $$X + Y \leq 0.5$$?

Accepted Answer

$$0.125$$

Question 18

Which of the following is a correct interpretation of the normalization axiom?

Accepted Answer

The probability of the entire sample space is 1.

Question 19

Which mathematical theory provides the foundation for probability spaces as measures with total mass 1?

Accepted Answer

Measure Theory

Question 20

In a discrete uniform probability law, the probability of an event is proportional to which property of the event?

Accepted Answer

Its cardinality

Foundations of Probability Theory: Sample Spaces, Events, and Probability Laws

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