# Event: Mutually Exclusive Events, Collectively Exhaustive Events, Independent Events, Simple and Compound Events

04/05/2021

Mutually Exclusive Events

When two events (call them “A” and “B”) are Mutually Exclusive it is impossible for them to happen together:

P(A and B) = 0

“The probability of A and B together equals 0 (impossible)”

But, for Mutually Exclusive events, the probability of A or B is the sum of the individual probabilities:

P(A or B) = P(A) + P(B)

“The probability of A or B equals the probability of A plus the probability of B”

Collectively Exhaustive Events

In probability, a set of events is collectively exhaustive if they cover all of the probability space: i.e., the probability of any one of them happening is 100%. If a set of statements is collectively exhaustive, we know at least one of them is true.

If you are rolling a six-sided die, the set of events {1, 2, 3, 4, 5, 6} is collectively exhaustive. Any roll must be represented by one of the set.

Sometimes a small change can make a set that is not collectively exhaustive into one that is. A random integer generated by a computer may be greater than or less than 5, but those are not collectively exhaustive options. Changing one option to “greater than or equal to five” or adding five as an option makes the set fit our criteria.

Another way to describe collectively exhaustive events is that their union must cover all the events within the entire sample space. For example, events A and B are said to be collectively exhaustive if

A U B=S

where S is the sample space.

Independent Events

Independent Events are not affected by previous events.

A coin does not “know” it came up heads before.

And each toss of a coin is a perfect isolated thing.

Probability of an event happening = Number of ways it can happen / Total number of outcomes

Simple and Compound Events

A simple event is one that can only happen in one way in other words, it has a single outcome. If we consider our previous example of tossing a coin: we get one outcome that is a head or a tail.

A compound event is more complex than a simple event, as it involves the probability of more than one outcome. Another way to view compound events is as a combination of two or more simple events.

Simple Event

An event that has a single point of the sample space is known as a simple event in probability.

Probability of an event occurring = No. of favorable outcomes / Total no. of outcomes

Compound Event

If an event has more than one sample point, it is termed as a compound event. The compound events are a little more complex than simple events. These events involve the probability of more than one event occurring together. The total probability of all the outcomes of a compound event is equal to 1.

To calculate probability, the following formula is used:

Probability of an event = [The number of favorable outcomes] / [the number of total outcomes].

First, we find the probability of each event occurring. Then we will multiply these probabilities together. In the case of a compound event, the numerator (number of favourable outcomes) will be greater than 1.

For example, the probability of rolling an odd number on a die, then tossing a head on a coin.

Here P(odd number) = 3/6