Independent Event - Practice Questions & MCQ

Two or more events are said to be independent if the occurrence or non-occurrence of any of them does not affect the probability of occurrence or non-occurrence of other events.
Two events $A$ and $B$ are said to be independent, if
1. $P(A \mid B)=P(A)$
2. $P(B \mid A)=P(B)$

A third result can also be obtained for independent events
From the multiplication rule of probability, we have

$
P(A \cap B)=P(A) P(B \mid A)
$
Now if $A$ and $B$ are independent, then $P(B \mid A)=P(B)$, so

$
\text { 3. } P(A \cap B)=P(A) \cdot P(B)
$

To show two events are independent, you must show only one of the above three conditions.
If two events are NOT independent, then we say that they are dependent.

With and Without replacement

In some questions, like the ones related to picking some object from a bag with different kinds of objects in it, objects may be picked with replacement or without replacement.

With replacement: If each object re-placed in the box after it is picked, then that object has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.

Without replacement: When sampling is done without replacement, then probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.

Difference between Independent events and Mutually Exclusive events

$A$ and $B$ are mutually exclusive events if they cannot occur at the same time. So, if $A$ occurred then $B$ cannot occur and vice versa. This means that $A$ and $B$ do not share any outcomes and $P(A \cap B)=0$. Also, $P(A / B)=0$ and $P(B / A)=0$.

But in case of independent events $A$ and $B, P(A / B)=P(A)$ [and not 0 as in case of mutually exclusive events], and $P(B / A)=P(B)$ [not 0]

Three Independent Events

Three events A, B and C are said to be mutually independent, if

$
\begin{aligned}
& P(A \cap B)=P(A) \cdot P(B) \\
& P(A \cap C)=P(A) \cdot P(C) \\
& P(B \cap C)=P(B) \cdot P(C) \\
& P(A \cap B \cap C)=P(A) \cdot P(B) \cdot P(C)
\end{aligned}
$

and

If at least one of the above is not true for three given events, we say that the events are not independent.

Properties of Independent Event

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

$
\text { 1. } \begin{aligned}
P(A \cup B) & =P(A)+P(B)-P(A \cap B) \\
& =P(A)+P(B)-P(A) \cdot P(B)
\end{aligned}
$

2. Event $A^{\prime}$ and $B$ are independent.

$
\begin{aligned}
P\left(A^{\prime} \cap B\right) & =P(B)-P(A \cap B) \\
& =P(B)-P(A) P(B) \\
& =P(B)(1-P(A)) \\
& =P(B) P\left(A^{\prime}\right)
\end{aligned}
$

3. Event A and $\mathrm{B}^{\prime}$ are independent.
4. Event $A^{\prime}$ and $B^{\prime}$ are independent.

$
\begin{aligned}
P\left(A^{\prime} \cap B^{\prime}\right) & =P\left((A \cup B)^{\prime}\right) \\
& =1-P(A \cup B) \\
& =1-P(A)-P(B)+P(A) \cdot P(B) \\
& =(1-P(A))(1-P(B)) \\
& =P\left(A^{\prime}\right) P\left(B^{\prime}\right)
\end{aligned}
$