Impossible Events

Introduction

The topic of probability discusses the possibility of things occurring and quantifies the same in numerical values. Therefore probability has several outcomes. These outcomes are known as events. There are several types of events in probability that exhibit what kind of outcome is possible. The possibilities that can occur are known as the sample space. The topic of probability is part of set theory. Set theory is the study of the properties of different groups and the mathematical expression of the same. 

The probability formula highlights the ratio between a favourable outcome and total outcomes. It is mathematically expressed as: Probability of an event = Favourable outcome/ Total Possible Outcomes. 

Types of Events

The following are types of events that can occur in probability. 

1. Simple Event

In this type of event, we can find only one single element in the sample space that represents the event. 

Example : In case of tossing a coin, E = the event of getting a head, F = the event of getting a tail. They are both elementary events.

In throwing a die, A is an event of getting 5. This is an elementary event. Meanwhile, B is the event of getting an even number. This is not an elementary event because the favourable outcomes for it are 2, 4, 6. 

It must be noted that the sum of probabilities of all elementary events is 1.

2. Compound Event 

This type of event occurs when there is more than one element in the sample space of the set representing an event. 

Example: When we throw a die, having S = {1, 2, 3, 4, 5, 6}, and an odd number is given by E = {1, 3, 5}.

3. Sure or Certain Event

Sure or certain events are events that occur at every experiment. To be precise, all outcomes of the experiment are favourable outcomes. 

Examples: When tossing a coin, the only 2 options available are a head or tail. 

Similarly, when throwing a die, the event of getting a natural number less than 7 is a sure event.

4. Impossible Event

An event that will surely not occur at any experiment is known as an impossible event.

Examples:

(i) Getting Seven in die throwing.

(ii) Getting a ‘Sum of 13 while throwing a pair of dice.

5. Equivalent Event or Identical Event

Equivalent or identical events are events when the occurrence of one event is based on the occurrence of the other and vice versa. 

Example: “even face” and “face-2” or “face-4” or “face-6” which are two identical events.

6. Equally Likely Event

When there is no other go expect the occurence of one event in preference to the other. This is known as equally likely events.

Example: When an unbiased coin is tossed the chances of getting ahead or a tail are the same.

7. Exhaustive Event

All the possible outcomes in any experiment are termed as exhaustive events. 

Example: If you throw a die, there occurs 6 exhaustive events in a trial.

8. Favourable Event

The total number of outcomes that show the occurrence of events in random experiments are called favourable events. 

Example: When you throw 2 dice, the probability of getting a sum of 5 is 4. The favourable outcomes are the pairs of (1, 4), (2, 3), (3, 2), and (4, 1).

9. Mutually Exclusive Event

If you find no common elements between two or more events, then it is termed as mutually exclusive events. 

Example: When you throw a die, the event of “even face” and “odd face” are mutually exclusive.

10. Finding the Probability of Impossible Event

It must be mentioned that the probability of getting an impossible event is always 0. This is because, for an impossible event, E = 0 and thus, P(E) = 0. 

This can be explained through an example. When finding the probability of any event, we must consider the total outcomes as well the number of favourable outcomes. When we roll a fair die, you will get a number not more than 6. There is a possibility of getting 1, 2, 3, 4, 5, or 6. But the outcome of getting 9 or any number greater than 6 is an impossible outcome, which cannot be included in sample space. Thus, it can be concluded that the probability of getting an impossible event is 0.

Conclusion

Possibility of things is used in several industries such as weather, sports, insurance, and games. Apart from uses, some common experiments used to explain probability are that of tossing a coin, rolling a die, and pulling cards from a deck of playing cards. As seen above, there are several types of events that occur in probability. They are based on the possible sample set. They include sure events, impossible events as well as simple and compound events. All these types of events are used as part of set theory and probability.