What Is the Multiplication Theorem of Probability?

The multiplication theorem of probability establishes a precise relationship between the probability of the intersection of events and conditional probabilities. This theorem is fundamental for determining the likelihood that multiple events occur simultaneously, especially when events are dependent or independent.

Mathematical Statement and Proof of the Multiplication Theorem of Probability

Consider two events $A$ and $B$ defined on the same sample space $S$, with $P(A) > 0$. The probability of both $A$ and $B$ occurring is denoted by $P(A \cap B)$. The conditional probability of $B$ given $A$ is defined as:

$P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}$

Multiplying both sides by $P(A)$ gives:

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

Therefore, the multiplication theorem of probability is stated as:

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

Similarly, if $P(B) > 0$, the conditional probability of $A$ given $B$ is:

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$

Multiplying both sides by $P(B)$ gives:

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

Thus, $P(A \cap B) = P(B) \cdot P(A \mid B)$ also holds.

For three events $A$, $B$, and $C$ with $P(A) > 0$ and $P(A \cap B) > 0$:

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

Special Case: Independent Events and Multiplication Rule

Events $A$ and $B$ are called independent if the occurrence of one does not affect the probability of the other, which mathematically means $P(B \mid A) = P(B)$. Substituting this into the multiplication theorem:

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

This form of the theorem simplifies calculation when the independence of events is established.

For three independent events $A$, $B$, and $C$:

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

Generalization: Multiplication Theorem of Probability for Three Events

For any three events $A$, $B$, and $C$ such that $P(A) > 0$ and $P(A \cap B) > 0$:

First, the probability of $A$ and $B$ and $C$ all occurring can be written as $P(A \cap B \cap C)$. The conditional probability $P(B \cap C \mid A)$ is:

$P(B \cap C \mid A) = \dfrac{P(A \cap B \cap C)}{P(A)}$

But $P(B \cap C \mid A) = P(B \mid A) \cdot P(C \mid A \cap B)$. Therefore:

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

This generalization aligns with the Fundamental Counting Principle applied to probabilities, where each event may conditionally depend on the occurrence of previous events.

Worked Examples Using the Multiplication Theorem of Probability

Consider a problem where two cards are drawn sequentially from a standard deck of $52$ cards without replacement. To determine the probability that both cards are kings:

Given: There are $4$ kings in a deck.

Step 1: Probability that the first card is a king: $P(\text{First King}) = \dfrac{4}{52}$.

Step 2: After one king is drawn, $3$ kings and $51$ cards remain. Probability the second card is a king given the first is king: $P(\text{Second King} \mid \text{First King}) = \dfrac{3}{51}$.

Step 3: By the multiplication theorem:

$P(\text{Both Kings}) = P(\text{First King}) \cdot P(\text{Second King} \mid \text{First King})$

$= \dfrac{4}{52} \cdot \dfrac{3}{51}$

$= \dfrac{12}{2652}$

$= \dfrac{1}{221}$

A similar approach applies when drawing balls sequentially from a bag or any scenario where prior outcomes influence subsequent probabilities. Observe that conditional probabilities change with each event since the sample space reduces.

To see the theorem for independent events, consider tossing two coins. The outcome of one coin does not affect the other.

Given: Probability the first coin is head: $P(\text{First Head}) = \dfrac{1}{2}$.

Probability the second coin is head: $P(\text{Second Head}) = \dfrac{1}{2}$.

By the multiplication theorem for independent events:

$P(\text{Both Heads}) = \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{1}{4}$

For further foundation on the concepts of probability, refer to Understanding Probability.

Comparison with Addition Rule in Probability Computation

To compute the probability of either event $A$ or $B$ occurring, the addition theorem in probability is applied:

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

Here, $P(A \cap B)$ is often calculated using the multiplication theorem.

Key Observations and Exam Cues on Error Patterns

Failure to correctly identify dependence or independence between events leads to erroneous probability computations. Always verify whether $P(B \mid A) = P(B)$ holds before applying the simplified multiplication rule.

In problems involving sequential actions without replacement, conditional probabilities must be recalculated after each action to account for the changed sample space.

The multiplication theorem allows construction of probabilities for complex dependent event structures in multi-stage experiments, as shown in problems involving drawing multiple objects or consecutive selections.

For additional exam practice using the multiplication theorem, review more examples under Multiplication Theorem Of Probability.