What are mutually Exclusive Events?
A mutually exclusive event occurs when two events cannot occur at the same time. For example, while tossing a coin, the coin landing either on heads or tails at one toss, is mutually exclusive.
What are Independent Events?
Two events are said to be independent if the occurrence of one event does not affect another event. For example, if a coin is tossed four times and ‘heads’ appear thrice, the fourth event is independent of the previous events.
What is the difference between Mutually Exclusive and Independent Events?
Mutually exclusive events are events where the occurrence of events is not simultaneous, while for independent events, one event has no control or effect over the outcome of the other or the next event.
There is no overlap of sets for mutually exclusive events, while the sets will overlap for independent events.
The mathematical formula for mutually exclusive events is expressed as P(A and B) = 0, while the mathematical formula for independent events is given as P(A and B) = P(A)P(B).
[Note: Here, P(X) represents the probability of the occurrence of event X.]
Solved Examples
Example 1: What is the probability of drawing a queen followed by a king in a pack of 52 cards with replacement? Also state whether the events are mutually exclusive or independent events.
Solution:
Given in the problem, there is an independent event condition.
The probability of drawing a queen in the first condition \(=\frac{4}{52}\)
The probability of drawing a king in the second condition after a queen with replacement \(=\frac{4}{52}\)
Hence, the probability of drawing a queen followed by a king \(=\frac{4}{52} \times \frac{4}{52} = \frac{16}{2704}=\frac{1}{169}\)
Here, the occurrence of the first event does not affect the occurrence of the second event. So, the events are independent events.
Hence, the probability of drawing a queen followed by a king with replacement is \(\frac{1}{169}\) .
Example 2: If a coin is tossed four times and ‘heads’ appear all four times, what is the probability that the next toss will be ‘heads’ as well?
Solution:
The given problem is an example of an independent event.
Hence, the chance of getting ‘heads’ in the next toss is \(\frac{1}{2}\) or 0.5 as each event is independent of the other.
Example 3: What is the probability of getting a ‘3’ or ‘5’ when rolling a dice?
Solution:
Here, we will find out the number of ways ‘3’ or ‘5’ can happen.
Number of ways it can happen = 2 (‘3’ and ‘5’)
Total number of outcomes = 6 (‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’)
Hence, the probability will be \(\frac{2}{6}=\frac{1}{3}\) or 0.33
An experiment is defined as a procedure that is related to several results or outcomes.
A collection of one or more outcomes for a certain experiment is known as an event.
Mutually exclusive events are events that cannot occur at the same time. Independent events are ‘independent’ of the probability of occurrence of the other. Hence, an event cannot be mutually exclusive and independent.
Experimental probability is defined as the probability that is dependent on repeated trials of a particular experiment.