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# Independence of events

schedule Aug 11, 2023
Last updated
local_offer
Probability and Statistics
Tags
Definition.

# Independent events

Two events are said to be independent if the occurrence (or non-occurrence) of one event does not affect the probability that the other event will occur.

Example.

## Coins toss as an example of independent events

Suppose we toss a coin twice and we define the following events:

• event $A$ - obtaining a heads for the first toss

• event $B$ - obtaining a heads for the second toss

Because the outcome of the first coin toss does not affect the outcome of the next coin toss, these events are independent.

Example.

## Relationship between outcome of football match and outcome of coin toss

Suppose we have the following events:

• event A - the weather is rainy

• event B - obtaining a heads for a single coin toss

Clearly, these events are independent because the weather has absolutely no impact on the probability of a tossed coin resulting in heads.

Theorem.

# Multiplication rule of independent events

If $A$ and $B$ are independent events, then:

$$\mathbb{P}(A\cap{B})= \mathbb{P}(A)\cdot\mathbb{P}(B)$$

Proof. Rearranging the formula for conditional probability:

$$\begin{equation}\label{eq:KNmyJfG7wjdt6LKwHg2} \mathbb{P}(A\cap{B})=\mathbb{P}(A|B)\cdot\mathbb{P}(B) \end{equation}$$

If events $A$ and $B$ are independent, then this means that probability of $A$ is not affected by the occurrence of $B$, which means that $\mathbb{P}(A\vert{B}) = \mathbb{P}(A)$. Therefore \eqref{eq:KNmyJfG7wjdt6LKwHg2} becomes:

$$\begin{equation}\label{eq:nIOEIGHvKWmtcy3pnJp} \mathbb{P}(A\cap{B})=\mathbb{P}(A)\cdot\mathbb{P}(B) \end{equation}$$
Example.

## Rolling two fair dices

Suppose we roll two fair dices. What is the probability that the outcome of both the dices is even?

Solution. Let events $A$ and $B$ represent the outcome of the first and second dice respectively. Events $A$ and $B$ are independent because the outcome of the first dice does not affect outcome of second dice. Therefore, the probability that the outcome of both dices is even is given by:

\begin{align*} \mathbb{P}(A\text{ is even }{\cap}\; B\text{ is even}) &=\mathbb{P}(A\text{ is even})\cdot\mathbb{P}(B\text{ is even})\\ &=\frac{3}{6}\cdot\frac{3}{6}\\ &=\frac{1}{4} \end{align*}

# Difference between independent events and disjoint events

There is a common misconception that independent events and disjoint events are equivalent. Disjoint events are events that cannot occur simultaneously, and are illustrated in the diagram below: For instance, suppose we toss a coin once and define the following two events:

• event $A$ - outcome of toss is heads.

• event $B$ - outcome of toss is tails.

Clearly, these events cannot happen at the same time - only one of the two events can occur and not both. Now, the question is, are events $A$ and $B$ independent? These two events are highly dependent because if we know the result of event $A$, then we will definitely know the result of event $B$.

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