# What are Independent Events?

Independent events are those events whose occurrence is not dependent on any other event. For example, if we flip a coin in the air and get the outcome as Head, then again if we flip the coin but this time we get the outcome as Tail. In both cases, the occurrence of both events is independent of each other.

Independent events are those events whose occurrence is not dependent on any other event. If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events.

Examples:

• Tossing a coin.

Here, Sample Space S = {H, T} and both H and T are independent events.

• Rolling a die.

Sample Space S = {1, 2, 3, 4, 5, 6}, all of these events are independent too.

Both of the above examples are simple events. Even compound events can be independent events. For example:

• Tossing a coin and rolling a die.

Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.

These events are independent because only one can occur at a time.

Consider an example of rolling a die. If A is the event ‘the number appearing is greater than 3’ and B be the event ‘the number appearing is a multiple of 3’, then

• P(A)= 3/6 = 1/2 here favorable outcomes are {4,5,6}
• P(B) = 2/6 = 1/3 here favorable outcomes are {3,6}

Also, A and B is the event ‘the number appearing is odd and a multiple of 3’ so that P(A ∩ B) = 1/6

• P(A│B) = P(A ∩ B)/ P(B) = (1/6)/(1/3) = 1/2
• P(A) = P(A│B) = 1/2, 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)

Note: 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)

Note: We can calculate the probability of two or more Independent events by multiplying