If probability of occurrence of an event A is not affected by occurrence of another event B, then A and B are said to be **independent events**. Consider an example of rolling a die. If A is the event ‘the number appearing is odd’ and B be the event ‘the number appearing is a multiple of 3’.

Then

P(A) = \( \frac 36 \) = \( \frac 12 \) and P(B) = \( \frac 26 \) = \( \frac 13 \)

Also A and B is the event ‘the number appearing is odd and a multiple of 3’ so that

P(A ∩ B) = \( \frac 16 \)

P(A│B) = \( \frac {P(A ∩ B)}{ P(B)} \)

= \( \frac {\frac 16}{\frac 13 } \) = \( \frac 12 \)

P(A) = P(A│B) = \( \frac 12 \), which implies that the occurrence of event B has not affected the probability of occurrence of the event A.

If A and B are independent events, then P(A│B) = P(A)

Using Multiplication rule of probability,

P(A ∩ B) = P(B) . P(A│B)

P(A ∩ B) = P(B) . P(A)

A and B are two events associated with the same random experiment, then A and B are known as independent events if P(A ∩ B) = P(B) . P(A)

### What are mutually exclusive events?

Two events A and B are said to be mutually exclusive events if they cannot occur at the same time. Mutually exclusive events never have an outcome in common.

**Differences between independent events and mutually exclusive events:**

Independent Events |
Mutually exclusive events |

They cannot be specified based on the outcome of a maiden trial. | They are independent of trials |

Can have common outcomes | Can never have common outcomes |

If A and B are two independent events, then
P(A ∩ B) = P(B) . P(A) |
If A and B are two mutually exclusive events, then
P(A ∩ B) = 0 |