Mathematics PROBABILITY- Event , Types of events
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### Topics Covered

star Event
star Occurrence of an event
star Impossible and Sure Events
star Simple Event
star Compound Event
star Complementary Event
star The Event ‘A or B’
star The Event ‘A and B’
star The Event ‘A but not B’
star Mutually exclusive events
star Exhaustive events

### Event

\color{fuchsia} {ul ★ "Event "}

Any subset E of a sample space S is called an event.

Consider the experiment of tossing a coin two times. An associated sample space is color(navy)(S = {HH, HT, TH, T T}.)

Now suppose that we are interested in those outcomes which correspond to the occurrence of exactly one head.

We find that HT and TH  are the only elements of S corresponding to the occurrence of this happening (event). These two elements form the set color(navy)(E = { HT, TH})

We know that the set E is a subset of the sample space S . Similarly, we find the following correspondence between events and subsets of S.

color(red)"Description of events "  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(red) " Corresponding subset of ‘S’"
Number of tails is exactly 2  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A = {T T}
Number of tails is atleast one  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = {HT, TH, T T }
Number of heads is atmost one  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ C = {HT, TH, T T}
Second toss is not head  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D = { HT, T T}
Number of tails is atmost two  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ S = {HH, HT, TH, T T}
Number of tails is more than two \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ φ

### Occurrence of an event

\color{red} ✍️  color(blue)("The event E of a sample space S is said to have occurred") if the outcome ω of the experiment is such that color(blue)(ω ∈ E.)

\color{red} ✍️  If the outcome ω is such that color(blue)(ω ∉ E), we say that color(blue)("the event E has not occurred.")

color(red)("Consider the experiment of throwing a die.")

Let E denotes the event color(blue)"“ a number less than 4 appears”."

If actually ‘1’ had appeared on the die then color(green)("we say that event E has occurred.")

As a matter of fact if outcomes are 2 or 3, we say that event E has occurred .