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.

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

Mutually exclusive events occur when two events, A and B, cannot happen at the same time. In other words, the chance of both events occurring together is zero, which can be written as:

\(\text{P(A and B)=0}\)

To find the probability that either event A or event B happens, we should add the probability of each event:

\(\text{P(A or B)=P(A)+P(B)}\)

The steps to calculate the probability of mutually exclusive events is given as;

Step 1: Identify the two events, A and B, ensuring they do not co-occur.

Step 2: Make sure that the chance of both events happening together is zero:

\(\text{P(A and B)=0}\)

Step 3: Use the formula for the probability that both event happens:

\(\text{P(A or B)=P(A)+P(B)}\)

Step 4: Determine the individual probabilities of events A and B.

Step 5: Add these probabilities to find the combined probability.

Step 6: Remember that this combined probability represents the chance of either event occurring, but not both at once.

This simple approach helps when figuring out outcomes where two possibilities exclude each other, like tossing a coin or rolling a die.

Mutually exclusive events cannot occur at the same time, so the probability of both events occurring together is always zero. Therefore, the probability of two mutually exclusive events A and B is defined as follows:

\(P(AB) = 0\)

We know that, 

\(P (A U B) = P(A) + P(B) - P ( A∩B)\)

Considering A and B are mutually exclusive events, we get, 

\(P (A U B) = P(A) + P(B)\)

For example, while we are tossing a single die, the events “rolling a 2” and “rolling a 5” are mutually exclusive because the die cannot show both numbers at once. The probability of either event A or event B occurring is the sum of their individual probabilities.

\(P (A U B) = P(A) + P(B)\)

\(\text{P (2 or 5)}= \frac {1}{6} +\frac {1}{6} \)

\(\text{P(2 or 5)}= \frac{2}{6} = \frac {1}{3}\)

Therefore, the probability of rolling either a 2 or a 5 is \(\frac{1}{3}.\)

We can use Venn diagrams to illustrate mutually exclusive events. When a set is represented by a circle in a Venn diagram, mutually exclusive events are shown as two circles that do not overlap, indicating that the sets have no elements in common, as depicted in the image below.

For non-mutually exclusive events: In the case of non-mutually exclusive events, the Venn diagram shows an overlap between the two sets, indicating that they share some common elements, as illustrated in the image below.

Conditional probability refers to the likelihood of event A occurring, assuming that event B has already taken place.

For two independent events A and B, the conditional probability of event B given that event A has occurred is represented by P(B∣A) and is defined by the following formula.

\(P(B|A) = \frac{P (A ∩ B)}{P(A)}\)

According to the multiplication rule, 

\(P (A ∩ B) = 0\)

Therefore, 

\(P(B|A) = \frac{0}{P(A)}\)

\(P(B|A) = 0\)

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:
  
  \(\text{P(A′) = 1 − P(A)}\)

• Teachers should help their students learn the concept effectively. It can be done using real-life examples, such as rolling a die or pulling cards from a deck.
   
• Students should remember that in a Venn diagram, circles do not overlap for mutually exclusive events, but they do for non-mutually exclusive events.
   
• Teachers can begin the lesson with a question like “Can they happen together?” The events that happen together are not mutually exclusive, and the events that cannot occur together are mutually exclusive.
   
• Parents can help their children by asking them to note their everyday choices. Give them options for dinner. Ask them if they want pasta or pizza, which is a mutually exclusive event. 
   
• Play dice games and card games with children to ask if two numbers can happen at once or if two different cards can be pulled out at once.
   
• Parents should encourage asking back to check whether they have understood the concept. Ask them why rolling a two and a five at the same time is not possible.
   
• Start solving problems using the probability formula early. Verify the formula using dice, cards, or spinners. The formula is given as:
  
  \(\text{P(A or B)}=P(A)+P(B)\)

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.