Business Statistics Chapter 06: Probability Theory

Practice questions covering basic concepts, event operations, probability approaches, counting techniques, and fundamental theorems as outlined in Business Statistics Chapter 06.

What is defined as a numerical measure of uncertainty ranging from $0$ to $1$?

Probability, where $0$ represents impossibility and $1$ represents certainty.

In probability theory, what is a 'Random Experiment'?

Any repeatable process with outcomes that cannot be predicted with certainty.

What is the 'Sample Space' ($S$) and what are its 'Sample Points'?

The Sample Space is the set of all possible outcomes of a random experiment, while Sample Points are the individual possible outcomes.

How is an 'Event' defined in relation to the sample space?

An event is any collection of outcomes of an experiment or any subset of the sample space.

What characterizes the 'Intersection of Events' ($A \cap B$)?

It is an event containing all elements that are common to both events $A$ and $B$, expressed as $\{x \, / \, x \in A \text{ and } x \in B\}$.

What is the 'Union of two events' ($A \cup B$)?

An event containing all elements that belong to 'A' or to 'B' or to both, expressed as $\{x \, / \, x \in A \text{ or } x \in B\}$.

Define the 'Difference of 2 events' ($A - B$).

An event consisting of elements of $A$ which do not belong to $B$, also denoted as $A \cap B'$, expressed as $\{x \, / \, x \in A \text{ and } x \notin B\}$.

What is the 'Complement of an event' ($A'$)?

An event consisting of all the elements of the sample space $S$ that are not elements of $A$, expressed as $\{x \, / \, x \in S \text{ and } x \notin A\}$.

When are events ($A_1, A_2, \dots, A_n$) considered 'Collectively Exhaustive'?

When their union covers the entire sample space: $A_1 \cup A_2 \cup \dots \cup A_n = S$.

What are 'Mutually Exclusive' or 'disjoint' events?

Events that cannot occur simultaneously in a single trial of an experiment ($A_1 \cap A_2 = \emptyset$).

What are 'Equally Likely Events'?

Events that have the same chance of occurring.

According to the 'Classical Approach', how is the probability of an event $P(A)$ calculated?

$P(A) = \frac{\text{Number of favourable cases}}{\text{Total number of equally likely cases}} = \frac{n(A)}{n(S)} = \frac{m}{n}$.

How is the 'Relative Frequency Approach' to probability defined?

It is the limit of the relative frequency ($m/n$) as the number of repetitions ($n$) approaches infinity ($P(A) = \lim_{n \to \infty} \frac{m}{n}$).

By what other names is 'Relative Frequency Probability' known?

Posterior probability and Empirical probability.

What characterizes the 'Subjective Approach' to probability?

Probability assigned by an individual based on available evidence, knowledge, experience, and personal beliefs.

What are the first two rules under the 'Axiomatic Approach' to probability?

1. $P(A) \ge 0$; 2. $P(S) = 1$. Additionally, $P(\emptyset) = 0$.

What is the 'Addition Law of Probability' for any two events $A$ and $B$?

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

What formula is used to calculate the probability of 'A' but not 'B' ($P(A \cap B')$)?

$P(A \cap B') = P(A) - P(A \cap B)$.

What is 'Joint Probability'?

The probability of the joint occurrence of two or more events, denoted by $P(A \cap B)$.

What is 'Marginal Probability'?

The probability of one event ignoring any information about other events, also known as unconditional probability.

What is the definition and formula for 'Conditional Probability' ($P(B|A)$) when $P(A) > 0$?

The probability that event $B$ occurs subject to the condition that $A$ has already occurred: $P(B|A) = \frac{P(A \cap B)}{P(A)}$.

What is the 'Multiplication Rule of Probability' derived from conditional probability?

$P(A \cap B) = P(A) \times P(B|A)$ or $P(A \cap B) = P(B) \times P(A|B)$.

Under what condition are two events $A$ and $B$ considered independent?

If $P(B|A) = P(B)$ or $P(A|B) = P(A)$, leading to the specialized multiplication rule: $P(A \cap B) = P(A) \times P(B)$.

In counting techniques, what is the 'Factorial' of $n$ ($n!$)?

The product of positive integers from $1$ to $n$ ($n! = 1 \times 2 \times 3 \times \dots \times n$). By definition, $0! = 1$ and $1! = 1$.

What is the formula for 'Combinations' ($_nC_r$)?

$_nC_r = \frac{n!}{(n - r)! \, r!}$ where $r \le n$.

What is the 'Total Probability Law' for a random event $B$ given mutually exclusive and exhaustive events $A_i$?

$P(B) = \sum_{i=1}^{n} P(A_i) \times P(B|A_i)$.

What is the purpose of 'Tree Diagrams' in probability?

To graphically represent the sample space for experiments with multiple stages, where each branch shows a possible outcome.

What is the formula for 'Bayes’ Theorem' to find the updated probability $P(A_i|B)$?

$P(A_i | B) = \frac{P(B | A_i) \times P(A_i)}{\sum P(B | A_i) \times P(A_i)}$.