Laws of probability | FE Mechanical Probability & statistics

Laws of probability

Probability quantifies the likelihood of events occurring, providing a numerical measure of uncertainty.

Sample space & events

Definitions

Sample space ( $S$ or $Ω$ ): The set of all possible outcomes of an experiment.
Event: A subset of the sample space (a collection of outcomes).
Simple event: An event consisting of a single outcome.
Compound event: An event consisting of multiple outcomes.

Axioms of probability

For any event $A$ in sample space $S$ :

Non-negativity: $0 \leq P (A) \leq 1$
Certainty: $P (S) = 1$
Additivity: If $A$ and $B$ are mutually exclusive (disjoint), then the probability of either $A$ and $B$ occurring is::

$P (A \cup B) = P (A) + P (B)$

Complement rule:

$P (\overset{ˉ}{A}) = 1 - P (A)$

Addition rule (general):

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

This may be thought of as removing the “double counted” intersection between $A$ and $B$ when the probabilities of $A$ and $B$ are added together. Note that for mutually exclusive events, $P (A \cap B) = 0$ , so $P (A \cup B) = P (A) + P (B)$ .

Conditional probability

The conditional probability of event $A$ given that event $B$ has occurred is:

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

provided $P (B) > 0$ .

Interpretation: This represents the probability of $A$ occurring in the reduced sample space where $B$ is known to have occurred.

Independent events

Events $A$ and $B$ are independent if the probability of $A$ occurring is unaffected by whether $B$ occurs:

$P (A ∣ B) = P (A)$

Equivalently, $A$ and $B$ are independent if:

$P (A \cap B) = P (A) \cdot P (B)$

Multiplication rule

The multiplication rule is:

For events in general (both independent and dependent events):

$P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)$

For independent events in particular, this reduces to:

$P (A \cap B) = P (A) \cdot P (B)$

Bayes’ theorem

Bayes’ theorem provides a way to update probabilities based on new information:

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$

More generally, if $A_{1}, A_{2}, \dots, A_{n}$ partition the sample space:

$P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Sample space & events

Sample space ( $S$ or $Ω$ ): all possible outcomes
Event: subset of sample space
- Simple event: single outcome
- Compound event: multiple outcomes

Axioms of probability

$0 \leq P (A) \leq 1$ (non-negativity)
$P (S) = 1$ (certainty)
Additivity for disjoint events: $P (A \cup B) = P (A) + P (B)$
Complement rule: $P (\overset{ˉ}{A}) = 1 - P (A)$
General addition rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Conditional probability

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ (if $P (B) > 0$ )
Probability of $A$ given $B$ has occurred

Independent events

$P (A ∣ B) = P (A)$ (independence definition)
$P (A \cap B) = P (A) \cdot P (B)$ (independent events)

Multiplication rule

General: $P (A \cap B) = P (A) \cdot P (B ∣ A)$
For independent events: $P (A \cap B) = P (A) \cdot P (B)$

Bayes’ Theorem

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$
For partitions: $P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Laws of probability

Probability quantifies the likelihood of events occurring, providing a numerical measure of uncertainty.

Sample space & events

Definitions

Sample space ( $S$ or $Ω$ ): The set of all possible outcomes of an experiment.
Event: A subset of the sample space (a collection of outcomes).
Simple event: An event consisting of a single outcome.
Compound event: An event consisting of multiple outcomes.

Axioms of probability

For any event $A$ in sample space $S$ :

Non-negativity: $0 \leq P (A) \leq 1$
Certainty: $P (S) = 1$
Additivity: If $A$ and $B$ are mutually exclusive (disjoint), then the probability of either $A$ and $B$ occurring is::

$P (A \cup B) = P (A) + P (B)$

Complement rule:

$P (\overset{ˉ}{A}) = 1 - P (A)$

Addition rule (general):

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

Conditional probability

The conditional probability of event $A$ given that event $B$ has occurred is:

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

provided $P (B) > 0$ .

Interpretation: This represents the probability of $A$ occurring in the reduced sample space where $B$ is known to have occurred.

Independent events

Events $A$ and $B$ are independent if the probability of $A$ occurring is unaffected by whether $B$ occurs:

$P (A ∣ B) = P (A)$

Equivalently, $A$ and $B$ are independent if:

$P (A \cap B) = P (A) \cdot P (B)$

Multiplication rule

The multiplication rule is:

For events in general (both independent and dependent events):

$P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)$

For independent events in particular, this reduces to:

$P (A \cap B) = P (A) \cdot P (B)$

Bayes’ theorem

Bayes’ theorem provides a way to update probabilities based on new information:

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$

More generally, if $A_{1}, A_{2}, \dots, A_{n}$ partition the sample space:

$P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Laws of probability

Probability quantifies the likelihood of events occurring, providing a numerical measure of uncertainty.

Sample space & events

Definitions

Sample space ( $S$ or $Ω$ ): The set of all possible outcomes of an experiment.
Event: A subset of the sample space (a collection of outcomes).
Simple event: An event consisting of a single outcome.
Compound event: An event consisting of multiple outcomes.

Axioms of probability

For any event $A$ in sample space $S$ :

Non-negativity: $0 \leq P (A) \leq 1$
Certainty: $P (S) = 1$
Additivity: If $A$ and $B$ are mutually exclusive (disjoint), then the probability of either $A$ and $B$ occurring is::

$P (A \cup B) = P (A) + P (B)$

Complement rule:

$P (\overset{ˉ}{A}) = 1 - P (A)$

Addition rule (general):

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

Conditional probability

The conditional probability of event $A$ given that event $B$ has occurred is:

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

provided $P (B) > 0$ .

Interpretation: This represents the probability of $A$ occurring in the reduced sample space where $B$ is known to have occurred.

Independent events

Events $A$ and $B$ are independent if the probability of $A$ occurring is unaffected by whether $B$ occurs:

$P (A ∣ B) = P (A)$

Equivalently, $A$ and $B$ are independent if:

$P (A \cap B) = P (A) \cdot P (B)$

Multiplication rule

The multiplication rule is:

For events in general (both independent and dependent events):

$P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)$

For independent events in particular, this reduces to:

$P (A \cap B) = P (A) \cdot P (B)$

Bayes’ theorem

Bayes’ theorem provides a way to update probabilities based on new information:

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$

More generally, if $A_{1}, A_{2}, \dots, A_{n}$ partition the sample space:

$P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Sample space & events

Sample space ( $S$ or $Ω$ ): all possible outcomes
Event: subset of sample space
- Simple event: single outcome
- Compound event: multiple outcomes

Axioms of probability

$0 \leq P (A) \leq 1$ (non-negativity)
$P (S) = 1$ (certainty)
Additivity for disjoint events: $P (A \cup B) = P (A) + P (B)$
Complement rule: $P (\overset{ˉ}{A}) = 1 - P (A)$
General addition rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Conditional probability

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ (if $P (B) > 0$ )
Probability of $A$ given $B$ has occurred

Independent events

$P (A ∣ B) = P (A)$ (independence definition)
$P (A \cap B) = P (A) \cdot P (B)$ (independent events)

Multiplication rule

General: $P (A \cap B) = P (A) \cdot P (B ∣ A)$
For independent events: $P (A \cap B) = P (A) \cdot P (B)$

Bayes’ Theorem

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$
For partitions: $P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Laws of probability

Probability quantifies the likelihood of events occurring, providing a numerical measure of uncertainty.

Sample space & events

Definitions

Sample space ( $S$ or $Ω$ ): The set of all possible outcomes of an experiment.
Event: A subset of the sample space (a collection of outcomes).
Simple event: An event consisting of a single outcome.
Compound event: An event consisting of multiple outcomes.

Axioms of probability

For any event $A$ in sample space $S$ :

Non-negativity: $0 \leq P (A) \leq 1$
Certainty: $P (S) = 1$
Additivity: If $A$ and $B$ are mutually exclusive (disjoint), then the probability of either $A$ and $B$ occurring is::

$P (A \cup B) = P (A) + P (B)$

Complement rule:

$P (\overset{ˉ}{A}) = 1 - P (A)$

Addition rule (general):

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

Conditional probability

The conditional probability of event $A$ given that event $B$ has occurred is:

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

provided $P (B) > 0$ .

Interpretation: This represents the probability of $A$ occurring in the reduced sample space where $B$ is known to have occurred.

Independent events

Events $A$ and $B$ are independent if the probability of $A$ occurring is unaffected by whether $B$ occurs:

$P (A ∣ B) = P (A)$

Equivalently, $A$ and $B$ are independent if:

$P (A \cap B) = P (A) \cdot P (B)$

Multiplication rule

The multiplication rule is:

For events in general (both independent and dependent events):

$P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)$

For independent events in particular, this reduces to:

$P (A \cap B) = P (A) \cdot P (B)$

Bayes’ theorem

Bayes’ theorem provides a way to update probabilities based on new information:

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$

More generally, if $A_{1}, A_{2}, \dots, A_{n}$ partition the sample space:

$P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$

Sample space & events

Sample space ( $S$ or $Ω$ ): all possible outcomes
Event: subset of sample space
- Simple event: single outcome
- Compound event: multiple outcomes

Axioms of probability

$0 \leq P (A) \leq 1$ (non-negativity)
$P (S) = 1$ (certainty)
Additivity for disjoint events: $P (A \cup B) = P (A) + P (B)$
Complement rule: $P (\overset{ˉ}{A}) = 1 - P (A)$
General addition rule: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Conditional probability

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ (if $P (B) > 0$ )
Probability of $A$ given $B$ has occurred

Independent events

$P (A ∣ B) = P (A)$ (independence definition)
$P (A \cap B) = P (A) \cdot P (B)$ (independent events)

Multiplication rule

General: $P (A \cap B) = P (A) \cdot P (B ∣ A)$
For independent events: $P (A \cap B) = P (A) \cdot P (B)$

Bayes’ Theorem

$P (A ∣ B) = \frac{P ( B ∣ A ) \cdot P ( A )}{P ( B )}$
For partitions: $P (A_{i} ∣ B) = \frac{P ( B ∣ A _{i} ) \cdot P ( A _{i} )}{\sum _{j = 1}^{n} P ( B ∣ A _{j} ) \cdot P ( A _{j} )}$