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What are mutually exclusive events?

  • Last Updated : 21 Sep, 2021

The word probability or chance is extremely frequently utilized in day-to-day life. For example, we generally say, ‘He may come today or ‘probably it may rain tomorrow’ or ‘most probably he will get through the examination’. All these phrases involve an element of uncertainly and probability is a concept which measures the uncertainties. The probability when defined in the simplest way is the chance of occurring a certain event when expressed quantitatively, i.e., the probability is a quantitative measure of the certainty. Probability also means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from 0 to at least 1.

The probability has its origin within the problems handling games of chance like gambling, coin tossing, die throwing, and playing cards. In all these cases the outcome of a trial is uncertain. These days probability is widely used in business and economies in the field of predictions for the future.

For example, once we toss a coin, either we get Head or Tail, only two possible outcomes are possible (H, T). But if we toss two coins within the air, there might be three possibilities of events to occur, like both the coins show heads or both show tails or one shows heads and one tail, i.e.(H, H), (H, T),(T, T).

Formula for Probability

The probability formula is defined because the possibility of an occasion to happen is adequate to the ratio of the number of favorable outcomes and therefore the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes

Some important terms and concepts

  • Random experiment or Trial: The performance of an experiment is called a trial. An experiment is characterized by the property that its observations under a given set of circumstances do not always lead to the same observed outcome but rather to the different outcomes. If in an experiment all the possible outcomes are known beforehand and none of the outcomes are often predicted with certainty, then such an experiment is named a random experiment.
  • Equally Likely Events: Events are said to be equally likely if there is no reason to accept anyone in preference to others. Thus, equally likely events mean the outcome is as likely to occur as the other outcome.
  • Simple and Compound Events: In the case of simple events we consider the probability of happening or non-happening of single events and in the case of compound events we consider the joint occurrence of two or more events.
  • Exhaustive Events: It is the total number of all possible outcomes of any trial.
  • Algebra of Events: If A and B are two events associated with sample space S, then
    • A ∪ B is that the event that either A or B or both occur.
    • A ∩ B is the event that A and B both occur simultaneously.
  • Mutually Exclusive Events: In an experiment, if the occurrence of an occasion precludes or rules out the happening of all the opposite events in the same experiment.
  • Probability of an Event: Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways. Then the probability of happening of the event or its success is expressed as;

P(E) = r/n

The probability that the event won’t occur or referred to as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event won’t occur.

Therefore,

we can say;

P(E) + P(E’) = 1

This means that the entire of all the possibilities in any random test or experiment is adequate to 1.

What are mutually exclusive events?

In an experiment, if the occurrence of an occasion precludes or rules out the happening of all the opposite events in the same experiment. Two events are said to be mutually exclusive if they can’t occur at an equivalent time or simultaneously. In other words, mutually exclusive events are called disjoint events. If two events are considered disjoint events, then the probability of both events occurring at an equivalent time is going to be zero.

Let’s understand with examples:

  • When a coin is tossed either head or tail will appear. Head and tail cannot appear simultaneously. Therefore, the occurrence of a head or a tail is two mutually exclusive events.
  • In throwing a die all the 6 faces numbered 1 to 6 are mutually exclusive since if any one of these faces comes, the possibility of others in the same trial is ruled out.

Sample Problems

Question 1. If P(A) = 0.20, P(B) = 0.35 and (P(A ∪ B)) = 0.51, are A and B mutually exclusive?

Answer:

Step 1: Adding up the possibilities of the separate events (A and B).

0.20 + 0.35 = 0.55

Step 2: Comparing answer to the given “union” statement (A ∪ B). If they’re an equivalent, the events are mutually exclusive. If they’re different, they’re not mutually exclusive. If they are mutually exclusive, then the union of the two events must be the sum of both, i.e. 0.20 + 0.35 = 0.55.

Therefore, 0.55 doesn’t equal 0.51 hence, the events aren’t mutually exclusive. 

Question 2. If P(A) = 0.30, P(B) = 0.55 and (P(A ∪ B)) = 0.85

e A and B mutually exclusive?

Answer:

Step 1: Adding up the possibilities of the separate events (A and B).

0.30 + 0.55 = 0.85

Step 2: Comparing answer to the given “union” statement (A ∪ B). If they’re an equivalent, the events are mutually exclusive. If they’re different, they’re not mutually exclusive. If they are mutually exclusive, then the union of the two events must be the sum of both, i.e. 0.30 + 0.55 = 0.85.

Therefore, 0.85 is equal to 0.85 hence, the events mutually exclusive events. 

Question 3. If P(A) = 0.10, P(B) = 0.15 and (P(A ∪ B)) = 0.22, are A and B mutually exclusive?

Answer:

Step 1: Adding up the possibilities of the separate events (A and B).

0.10 + 0.15 = 0.25

Step 2: Comparing answer to the given “union” statement (A ∪ B). If they’re an equivalent, the events are mutually exclusive. If they’re different, they’re not mutually exclusive. If they are mutually exclusive, then the union of the two events must be the sum of both, i.e. 0.10 + 0.15 = 0.25.

Therefore, 0.25 doesn’t equal 0.22 hence, the events aren’t mutually exclusive. 

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