Mutually Exclusive Events

Mutually exclusive events are two or more events that cannot occur at the same time or in the same trial. In mathematics, mutually exclusive events mean that the intersection of these events is an empty set. When one event is happening, the second event cannot happen.

In probability, we express this as:

P(A ∩ B) = 0

Where

• A and B are mutually exclusive events

• P(A and B) indicates the probability of both the events happening at the same time.

Take a look at the following steps to calculate mutually exclusive events:

• Step 1: Examine the sample space (all possible outcomes)
   
• Step 2: Take a look at the individual events
   
• Step 3: Check for any possible overlaps 
   
• Step 4: The events are mutually exclusive if there are no overlaps.

There are a lot of differences between mutually exclusive and independent events. Some of the differences are mentioned below:

Some of the rules of mutually exclusive events are mentioned below:

• Definition Rule: Mutually exclusive events are exclusive if the occurrence of one event prevents the other from happening.

• Union Rule:  When two events are mutually exclusive, the probability of either of them happening is equal to the sum of their individual probabilities. The formula is:

P(A ∪ B) = P(A) + P(B)

• Intersection Rule: For mutually exclusive events, the intersection is always zero because the events cannot occur at the same time. The formula is:

P(A ∩ B) = 0

• Complement Rule: The complement rule states that the probability of an event not occurring is 1 minus the probability of the event occurring. The formula is:

P(A′) = 1 − P(A)

There are many uses of mutually exclusive events. Let us now see the uses and applications of mutually exclusive events in different fields:

Medical Diagnosis: Mutually exclusive events are used in medical diagnoses, where specific diseases cannot occur simultaneously. If a patient is diagnosed with a particular disease, then they cannot have another mutually exclusive condition at the same time.

Weather Forecasting: In meteorology, predicting specific weather events like rain or sunshine are mutually exclusive. It either rains or does not at any given time or place.

Traffic Signals: When managing the traffic at intersections, green and red signals are mutually exclusive. If the signal is green, then it cannot be red at the same time.