What Are Events?

The result or the outcome of a random experiment are called events connected with the experiments. Thus, "head" and "tail" are the results of the random experiment of tossing an unbiased coin and are events connected with this experiment. Similarly, the outcomes of the random experiment of throwing an unbiased die from a box are 1, 2, 3, ....., 6 and each outcome is an event connected with this experiment. Similarly, the outcome is an event connected with this experiment. Usually, capital letters A, B, C, etc are used to denote events connected with a random experiment. Events are of the following types:

1) Compound or Composite Event

Let A and B be the events 'even face' and 'multiple of three' respectively in the random experiment of throwing an unbiased die. Clearly, the event A occurs when the result of the experiment is 'two' or 'four' or 'six': similarly, the event B occurs if the outcome of the experiment is 'three' or 'six'. Thus, event A can be decomposed into the events 'two', 'four', 'six'. Thus event A can be 'three' and 'six'. Such events are called compound events or Composite events.

2) Simple or Elementary Event

On the contrary, events 'one', 'two', etc connected with the same experiments cannot be composed further: such events are called Simple or Elementary events. Thus, decomposed, whereas the compound event is the outcome of a random experiment that can be decomposed further into simple events.

3) Mutually Exclusive Events

Two events A and B connected with a random experiment E, are said to be mutually exclusive if they cannot occur simultaneously. Symbolically events A and B are mutually exclusive when A union B is equal to null or impossible events. Two simple events connected with a random experiment are always mutually exclusive but two compound events may or may not be so. Let A, B, and C be the events 'even face', 'odd face', and 'a multiple of three' respectively in the random experiment throwing an unbiased die. Clearly, the events A and B cannot occur simultaneously, and hence, they are mutually exclusive: but the events B and C occur simultaneously if the result of the experiment is three and hence, they are not mutually exclusive.

4) Impossible and Certain (or sue) Events

In any random experiment we can think of an event that is logically impossible i.e., cannot occur at any performance of the experiment. Such an event is called an impossible event. For example, the event of drawing a black ball from a bag containing three red and four white balls is an impossible event. Similarly, the event 'seven' can never occur in the random experiment of throwing a die and is, therefore, an impossible event. An impossible event is denoted by f and the probability of occurrence of an impossible event is zero. Again, in any random experiment, we can think of an event that is sure to occur at every performance of the experiment. Such an event will be called a certain (or sure) event. Let A denote the event 'Head or Tail' in the random experiment of tossing an unbiased coin. Clearly, the event A is sure to occur at every performance of the experiment and is, therefore, a certain event. A certain event is usually denoted by S and the probability of occurrence of a certain event is one.

5) Equally Likely Events

Two or more events are said to e equally likely if, after taking into consideration all relevant evidence, none can be expected in preference to another. Simple events connected with a random experiment are always equally likely but compound events may or may not be so. Thus the simple events 'Head' and 'Tail' connected with the random experiment of tossing an unbiased coin are equally likely. Similarly, the six simple events 'one', 'two', ...., 'six' connected with the random experiment of throwing an unbiased die are equally likely but the compound events A and B are equally likely when P(A) = P(B).

Solved Examples

Question 1: Two unbiased dice are rolled together. Find the odds in favor of getting 2 digits, the sum of which is 7.

Solution: Evidently the first die may have 6 different outcomes, each of which can be associated with 6 different outcomes of the second die. Therefore, the sample space of the random experiment of throwing two unbiased dice together contains 6 x 6 = 36 equally likely event points. Let A denote the event that the sum of the digits in the two dice is 7. Clearly, event A contains 6 equally likely events viz., (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

Therefore, by the classical definition of probability, we get, P(A) = 6/36 = ⅙

Therefore, odds in favor of events A are 1:(6-1) = 1:5

Question 2: the odds in favor of an event are 4:3. The odds against another independent event are 2:3. What is the probability that at least one of the events will occur?

Solution: Assume that the given events are A and B.

Then by the problem, probability of occurrence of A = P(A) = \[\frac{4}{4+3}\] = \[\frac{4}{7}\]

And probability of occurrence of B = P(B) = \[\frac{3}{2+3}\] = \[\frac{3}{5}\]

Therefore, the probability of occurrence of at least one of the events A and B

= P(A⋃B) = P(A) + P(B) - P(A⋂B)

= P(A) + P(B) - P(A).P(B)

= \[\frac{4}{7}\] + \[\frac{3}{5}\] - \[\frac{4}{7}\].\[\frac{3}{5}\] = \[\frac{20+21-12}{35}\] = \[\frac{29}{35}\]

FAQ (Frequently Asked Questions)

Q1. What are the Exhaustive Events?

Ans. A set of events connected with a random experiment is said to be exhaustive if least one of the sets is sure to occur at every performance of the experiment. Simple events connected with a random experiment always constitute an exhaustive set of events. Consider the random experiment of throwing an unbiased die from a box.

let A1, A2,...A6 be the events 'one','two',...'six’ respectively. Clearly, at least one of these events will occur at every performance of the experiment and hence, they form an exhaustive set of events. In the same experiment, let A, B, and C be the events 'even face', 'multiple of three' and experiment, let A, B, and C be the events 'even face', 'multiple of three', and 'fie' respectively. Obviously none of the events A, B, or C occurs when the outcome of the experiment is 'one'; because at least one of these four events must necessarily occur at every performance of the experiment.

Q2. What are the Complementary Events?

Ans. Let A be an event connected with a random experiment E. The event of non-occurrence of A is called the event complementary to A and is denoted by A'. If A is the event of drawing a spade from a well-shuffled pack of 52 cards then the event complementary to A is the event of not drawing a spade from the same pack. The complementary event of getting an even face in the random experiment of throwing a die is the event of not getting an even face (i.e., the event of getting an odd face).