Events and their Types in Probability

In the theory of probability, an event is defined as a set of outcomes of an experiment i.e. it is the subset of the respective sample space of an experiment. A sample space is the entire possible set of outcomes of a random experiment. The likelihood of occurrence of an event is known as probability, which lies anywhere between 0 and 1. A typical definite outcome is an element of many different events whereas an experiment always includes a very different group of outcomes, the different events are usually not equal in nature. When the sample space is finite, any subset of the sample space is an event but if the sample space is infinite, this may not be the case. When defining a probability space, it is often required to exclude certain subsets of the sample space from being considered as even.

When three coins are simultaneously tossed, the sample space is represented as:

S = {(H, H, H) , (H, H, T) , (H, T, H) , (H, T, T ) , (T, H, H ) , (T, H, T) , (T, T, H) ,(T,T,T)}

And the set of all such possibilities. if we want to find solely the outcomes which have at least two heads, it may be given as:

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Hence, an event can be considered as a subset of the sample space, i.e., E is a subset of S. There could be a lot of events associated with a given sample space but for an element to be considered as an Event of the set of events E, it must be a subset of the outcomes of the experiment. Events may be found in any real-life random experiments like rolling a dice, tossing a coin, etc.

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Bernoulli trial

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Together an event and its complementary set are defined as the Bernoulli trial i.e. did the event occur or not? Some key points of this trial are-

1. Number of trials is finite.
2. Trials are independent of each other.
3. Trials have exactly 2 outcomes: success or failure.
4. The success probability remains almost unchanged in each trial.

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Probability of Occurrence of an Event

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The probability of occurrence of any event is described as the number of favorable outcomes to the total number of outcomes in an experiment. An event that’s certain to happen has a probability equal to 1. An event that can’t possibly happen has zero probability. If there is a certain chance that the event will happen, then the probability of the event is between 0 and 1.

P(E) = Number of Favourable Outcomes/Total Number of Outcomes

Odds of an Event

- Odds in favor: It is the ratio of no. of ways an outcome can occur to no. of ways it cannot occur.

- Odds against: It is the ratio of no. of ways an outcome cannot occur to the no. of ways it can occur.

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Types of Events in Probability

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• Impossible and sure events.
• Simple Events.
• Compound Events.
• Independent and Dependent Events.
• Equally likely Events.
• Mutually Exclusive Events.
• Exhaustive Events.
• Complementary Events.
• Events Associated with “OR”.
• Events Associated with “AND”.
• Event A but not B.

• Impossible and Sure Events

If the probability of occurrence of an event is 0, i.e. there is no chance of this event ever occurring, such an event is called an impossible event and if the probability of occurrence of an event is 1, i.e. is sure to occur in any given experiment it is called a sure/certain event.

• Simple Events

Any event consisting of a single point of the sample space is known as a simple/elementary event in probability. It’s an event that has exactly one outcome.

• Compound Event

As opposed to a simple event, if any event consists of more than one single point of the sample space then such an event is called a compound event. It involves combining together two or more events and thus finding the probability of such a combination of events.

• Independent Events and Dependent Events

If the occurrence of any event is totally unaffected by the occurrence of some other event, such events are called independent events in probability and the events that are affected by other events are called dependent events.

• Equally likely Events

When the resulting outcomes of an experiment are equally likely to happen, they are known as equally likely events. For instance, during a coin toss, you are equally likely to get either heads or tails.

• Mutually Exclusive Events

If the occurrence of any one event excludes the occurrence of another i.e. when both two events cannot occur at the same time and must always have a different outcome, such events are mutually exclusive events i.e. two events don’t have any common point. Two simple events are mutually exclusive, whereas two compound events may or may not be so.

• Exhaustive Events

A set of events where all the events together consume the entire sample space is called exhaustive events.

• Complementary Events

For any particular event (E), the non-occurrence of the event is called its complementary event, which is the existence of another event that represents the remaining elements of the sample space. These are the events that cannot occur at the same time. So when a dice is thrown, getting an odd face and an even face are complementary events.

• Events Associated with “OR”

If two events A and B are associated with ‘OR’ then it means that either A or B or both. The union symbol (∪) is used to represent OR in probability, denoted by-

AUB.

• Events Associated with “AND”

If two events A and B are associated with ‘AND’ then it means the intersection of elements that is common to both the events. The intersection symbol (∩) is used to represent AND in probability, denoted by-

A∩B.

• Event A but not B

It represents the difference between both events. Event A but not B represents all the outcomes that are present in A but not in B. It is denoted by-

A-B.

• Conditional Probability

If both E1 and E2 are two events that are associated with the sample space, then the conditional probability of the event E1 given that E2 has already occurred, for any random experiment, is given by-

P(E1/E2)= P(E1∩E2)/P(E2), P(E2)≠0

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Probability

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It is the measure of uncertainty of any event.

Probability (P) = No. of favourable outcomes/Total no. of outcomes.

Important Formulas

For any Event A,

1. 0 ≤ P (A) ≤ 1
2. P (A’) = 1 – P (A)
3. P (AUA’) = 1
4. P (A∩A’) = 0

For any two Events A and B,

1. P(AUB) + P(A∩B) = P(A)+P(B)
2. P (A-B) = P(A) - P (A∩B)

If B is subset of A then,

1. P(A) ≥ P(B)
2. P (A-B) = P(A) – P(B)