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 .