QCM : Introduction to Probability and Uncertainty — 10 questions

Questions et réponses du QCM

1. What does a probability function do in the measure-based view of probability?

It lists every outcome in the sample space
It assigns each event a real number between 0 and 1
It assigns probabilities only to the most likely event
It maps each outcome to a single real-valued measurement

It assigns each event a real number between 0 and 1

Explication

A probability function takes events as inputs and returns a number in the interval [0,1]. Its domain is the power set of the sample space, not just a single outcome.

2. What is the primary role of a probability function in the context of a sample space?

It assigns a probability to each event, which is a subset of the sample space.
It measures the frequency of outcomes over repeated trials.
It assigns a numerical size to each outcome in the sample space.
It determines the likelihood of a single outcome occurring in an experiment.

It assigns a probability to each event, which is a subset of the sample space.

Explication

The probability function assigns a probability to each event, which is a subset of the sample space, and its domain is the power set of the sample space. This formalizes the mathematical foundation of probability.

3. Why must the domain of a probability function include the empty set and the full sample space?

So that probabilities can exceed 1 for certain events
So that the sample space becomes countably infinite
So that the two extreme events can also receive probabilities
So that every outcome is assigned exactly one event

So that the two extreme events can also receive probabilities

Explication

Including both the empty set and the whole sample space allows probabilities to be assigned to the extreme events of impossibility and certainty. This fits the function-based definition of probability over all subsets of the sample space.

4. What is the domain of a probability function as described in the course outline?

The set of all outcomes in the sample space
The set of all possible events
The power set of the sample space
The set of all real numbers between 0 and 1

The power set of the sample space

Explication

The probability function's domain is the power set of the sample space, 2^S, which includes all possible subsets (events). This allows assigning probabilities to every event.

5. Which statement best describes Knightian uncertainty?

The possible outcomes are known, but their probabilities are not precisely known
The probabilities are known, but the sample space is empty
The outcomes and probabilities are both fully known
The possible outcomes are not known in advance, but probabilities are exact

The possible outcomes are known, but their probabilities are not precisely known

Explication

Knightian uncertainty refers to cases where the outcomes exist but their precise probabilities are not specified. This differs from radical uncertainty, where the set of possible outcomes is not completely known.

6. What is the primary purpose of a probability function in the context of a sample space?

To determine the most likely outcome of a random experiment.
To assign probabilities to each event as a subset of the sample space.
To measure the uncertainty associated with outcomes.
To assign a numerical size to each outcome in the sample space.

To assign probabilities to each event as a subset of the sample space.

Explication

The probability function's main purpose is to assign a probability value between 0 and 1 to each event (subset of the sample space), reflecting the likelihood of that event occurring.

7. What is the key feature of radical uncertainty?

Each outcome has equal probability by construction
The experiment can be repeated under identical conditions
The probabilities of known outcomes are not exactly measurable
The set of possible outcomes is not completely known

The set of possible outcomes is not completely known

Explication

Radical uncertainty means the outcome space itself is incomplete or unknown. The other choices describe different ideas, such as Knightian uncertainty or a uniform model.

8. When was the concept of probability distributions and cumulative distribution functions (CDFs) formally established as fundamental tools in the study of random variables?

In the 19th century with the advent of statistical mechanics
In the early 20th century with the formalization of measure theory
In the 17th century during the development of classical probability theory
In the late 20th century with the rise of computational statistics

In the early 20th century with the formalization of measure theory

Explication

Probability distributions and CDFs were formally established as fundamental tools in the early 20th century, notably with the development of measure theory which provided a rigorous mathematical foundation for probability theory.

9. How does a probability distribution differ from a cumulative distribution function (CDF) in describing a random variable?

A probability distribution assigns probabilities to each possible value of the variable, while the CDF gives the probability that the variable is less than or equal to a specific value.
The probability distribution is only used for continuous variables, whereas the CDF is used for discrete variables.
The probability distribution and the CDF are essentially the same, with the distribution being the derivative of the CDF.
A probability distribution provides the cumulative probability up to a point, whereas the CDF assigns probabilities to individual values.

A probability distribution assigns probabilities to each possible value of the variable, while the CDF gives the probability that the variable is less than or equal to a specific value.

Explication

A probability distribution assigns probabilities to each possible value of the random variable, while the CDF gives the probability that the variable is less than or equal to a specific value, thus serving different but related roles in describing the distribution.

10. Who is credited with formalizing the concept of the uniform distribution as a distribution where all outcomes are equally probable?

Pierre-Simon Laplace
Jacob Bernoulli
Andrey Kolmogorov
Leonard Euler

Jacob Bernoulli

Explication

Jacob Bernoulli is credited with early formalization of the uniform distribution concept, emphasizing that all outcomes in its support are equiprobable. The other options contributed to probability theory but are not specifically associated with the formalization of the uniform distribution.

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Probability — measure?

Assigns numerical size to events.

Sample space S

All possible outcomes of experiment.

Radical uncertainty — difference?

Outcomes are not fully known.

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