Impossible and Certain Events

To know what impossible and certain events are, let us first learn about what probability and events actually mean. Probability is defined as the ratio of the number of events in an entire set of equally probable outcomes that result in a given occurrence to the entire number of conceivable outcomes. In general, the probability of an event is a number between 0 and 1, with 0 signifying impossibility and 1 indicating certainty. The greater the probability of something happening, the more likely it is to happen. The word probability comes from the Latin word, “probabilitas”.

 Probability of Events

 The ratio of the number of favourable outcomes to the total number of outcomes of an event is known as a probability in Maths. The number of favourable outcomes can be expressed as ‘x’ in an experiment with the ‘n’ number of outcomes. Let us now look at the formula for calculating the probability of an event:

All possible results of an experiment are represented by the sample space. As a result, in probability, occurrences can be thought of as subsets of the sample. In probability, there are various kinds of events, each having its characteristics. This probability classification aids in the simplification of mathematical calculations.

Let us learn about impossible and certain events separately.

Impossible Events

An impossible event’s probability is always zero, as E = 0 for an impossible occurrence, and hence P(E) = 0.Probability is defined as the likelihood of an event occurring, which is computed as the ratio of the number of positive outcomes to the total number of event outcomes.

 In other words, probability is a numerical representation of the likelihood of an event occurring. We are constantly faced with real-world situations in which we must predict the outcome of an event.

We may be confident or suspicious about an event’s outcome. As the name suggests, a specific event has been guaranteed to take place. When a cricket player tosses a coin before a match, for example, he/she will select either to bat or bowl, which is known as a certain event.

As a result, the probability of a particular event is 1 while the probability of an impossible event is 0.

 Because it cannot happen under any circumstances, an impossible event has a chance of zero. Probability is a number that ranges from 0 to 1 and is expressed as a percentage. A value of 0 denotes an improbable condition, while a value of 1 denotes a certain occurrence.

 Take this, for example, the chances of pulling an orange ball from a set of red balls are nil, as receiving an orange ball when the set only contains red balls is an impossible event.

Impossible Event and Empty Set

An impossible event is identical to an empty set  ∅. This is a probability rule that can be expressed mathematically as p (∅ ) = 0. Although it can be beneficial in some calculations, the impossible event, which is based on a set with no items in this form, does not match any experimental outcome.

Even though the probability of an impossible event is zero, this does not imply that all events with a probability of zero are impossible. There are numerous cases in which an event does not occur under certain conditions (i.e., it has a probability of zero for a specific model or situation), but does occur under others.

Certain Events

A certain event is guaranteed to occur. If and only if P(E) = 1, E is a certain occurrence.

A specific event would be getting a head or a tail while flipping a coin once. Certain events are the ones whose occurrence is certain. The probability of an event like this will always be one. When rolling a dice, the likelihood of getting a number between 1 and 6 is a known event. As a result, the probability of a certain event occurring is one.

If the probability of an event occurring is zero, it is called an impossible event. If the chance of an event occurring is 1, it is referred to as a certain event. Alternatively, the empty set ∅ is an impossible event, while the sample space S is a certain event.

Conclusion

To summarise what we have learned till now, we know that If the probability of an event occurring is zero, it is called an impossible event. If the chance of an event occurring is 1, it is referred to as a certain event. The empty set ∅, on the other hand, is an impossible event, while the sample space S is a certain event.

However, even though the probability of an impossible event is zero, this does not imply that all events with a probability of zero are impossible. There are numerous cases in which an event does not occur under certain conditions (i.e., it has a probability of zero for a specific model or situation), but does occur under others. Here’s a simple illustration: What is the probability of selecting a girl student  at random from a classroom full of boys? Because it’s impossible to choose a girl in a classroom full of boys, the probability is zero. However, this does not rule out the possibility of choosing a girl if the circumstances are altered significantly (for example, by adding girls to the classroom).