QCM : Introduction to Business Probability — 14 questions

Questions et réponses du QCM

1. Which statement best describes the axiom of non-negativity in probability?

The probabilities of all events must add up to 1
Every event must have a probability greater than or equal to 0
Every event must have a probability less than 1
Only mutually exclusive events can have probabilities

Every event must have a probability greater than or equal to 0

Explication

Non-negativity requires that probabilities cannot be negative. The normalization axiom is the one that sets the total probability of the sample space equal to 1.

2. What is subjective probability based on?

Long-run relative frequencies from repeated trials
A person's information or beliefs
A fixed count of observed outcomes only
A requirement that every event has equal probability

A person's information or beliefs

Explication

Subjective probability comes from a person's information or beliefs about an event. It can differ between people because their information may differ.

3. What best describes a random experiment?

A single observed result from one run of a process
A rule for assigning probabilities to outcomes
A collection of all outcomes that can occur
A repeatable process with known possible outcomes but uncertain results

A repeatable process with known possible outcomes but uncertain results

Explication

A random experiment is a repeatable process where the set of possible outcomes is known, but the actual result is uncertain before the trial. A single observed result is a trial, not the experiment itself.

4. What does linearity of expected value allow you to do?

Assume the expected value must equal one observed outcome
Replace every probability weight with the same value
Ignore coefficients when combining random variables
Compute the expectation of a sum from the expectations of simpler parts

Compute the expectation of a sum from the expectations of simpler parts

Explication

Linearity means the expected value of a sum can be found by adding the expected values of the components, which simplifies calculations. It does not mean probabilities are all equal or that one outcome must occur.

5. What is a sample space?

The set of outcomes that matter for a particular question
The collection of all possible outcomes of a random experiment
A region in a Venn diagram where events overlap
One specific observed result from a trial

The collection of all possible outcomes of a random experiment

Explication

The sample space is the full collection of all possible outcomes. An event is the subset of outcomes relevant to a question, not the entire space.

6. What does the overlap region in a Venn diagram represent?

Outcomes that belong to neither event
All outcomes in the sample space
Outcomes that belong to both events
Outcomes that occur only once

Outcomes that belong to both events

Explication

The overlap region shows the outcomes shared by both events. This is the part that can cause double-counting if events are not disjoint.

7. Why is frequentist probability difficult to apply to some business situations?

Because it relies on long-run repetition of the same process
Because it cannot be used for repeated trials
Because it is based on personal beliefs instead of data
Because it requires events to be unique or one-time only

Because it relies on long-run repetition of the same process

Explication

Frequentist probability estimates probability from long-run relative frequencies, so it needs repeated trials of the same process. That makes it hard to use for essentially unique events like starting a new company.

8. Why are exhaustive events useful in probability modeling?

They ensure that at least one considered event occurs in every trial
They guarantee the events never overlap
They remove the need for a sample space
They make all probabilities equal

They ensure that at least one considered event occurs in every trial

Explication

Exhaustive events cover all possibilities, so something in the set must occur each trial. They may still overlap, so they are not necessarily mutually exclusive.

9. What is the expected value of a random variable?

The most likely outcome in a single trial
The value that must occur in the next trial
The weighted-average value across all possible outcomes
The total of all possible outcomes with no weights

The weighted-average value across all possible outcomes

Explication

Expected value is the weighted average of all possible outcomes, usually using probabilities as weights. It is a theoretical average, not necessarily a result from any one trial.

10. In a store-opening example, which statement correctly identifies the trial?

The number of customers on a given day
The uncertainty about whether customers will arrive
The set of all possible customer counts
Opening the store every morning

The number of customers on a given day

Explication

A trial is one single execution of the random experiment, so the number of customers on a given day is the observed result. Opening the store every morning is the experiment.

11. When do Venn diagrams become less practical?

When there are many events or high-dimensional relationships
When the probabilities sum to 1
When the events are mutually exclusive
When the sample space is fixed before observing outcomes

When there are many events or high-dimensional relationships

Explication

Venn diagrams are helpful for visualizing a small number of events, but they become cumbersome with many events or complex relationships. The other choices do not describe a limitation of Venn diagrams.

12. Why can the empty set be considered an event?

Because it represents an impossible event
Because it represents the entire sample space
Because it is a single observed outcome
Because it contains every possible outcome

Because it represents an impossible event

Explication

The empty set is an event because it corresponds to an impossible event, meaning no outcomes satisfy it. It is not the same as the sample space or a single outcome.

13. Which statement best describes mutually exclusive events?

They can only occur after repeated trials
They must cover every possible outcome
They cannot occur at the same time in the same trial
They are guaranteed to overlap in some outcomes

They cannot occur at the same time in the same trial

Explication

Mutually exclusive events have no overlap, so they cannot both happen in the same trial. This is what prevents double-counting when adding their probabilities.

14. Why are probability axioms needed as the starting point of probability theory?

They make every event equally likely by definition
They eliminate the need to define a sample space
They allow probabilities to be calculated only from past observations
They provide a consistent framework that prevents impossible probability values

They provide a consistent framework that prevents impossible probability values

Explication

The axioms establish a consistent mathematical foundation so probabilities stay meaningful and do not become negative or greater than 1. They do not force equal likelihood or replace the sample space.

Révisez avec les flashcards

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Random experiment — definition?

A repeatable process with known outcomes.

Trial — role?

Single execution producing one result.

Sample space — what?

All possible outcomes of an experiment.

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