## Sample space

A sample space is the set of all possible outcomes of a random experiment. We denote it by S, and the total number of elements in the sample is the size of the sample space and we denote it by n(S).

Discrete sample space- Sample space in which sample points are finite or countably infinite.

For example-

If we throw a dice, then the spl space is

S = {1, 2, 3, 4, 5, 6} and n(S) = 6.

If we toss a coin twice or simultaneously then-

S = {HH, HT, TH, TT}

If toss a coin 4 times or toss four coins simultaneously then-

S = {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, HTTH, THHT,

THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT} and n(S) = 16.

Note-If a random experiment with x possible outcomes is performed n times, then the total number of elements in the sample is x^n

## Events and their types

1. Exhaustive Events or Sample Space: The set of all possible outcomes of a single performance of an experiment is exhaustive events or sample space. Each outcome is a sample point.

In case of tossing a coin once, S = (H, T) is the sample space. Two outcomes – Head and Tail

– constitute an exhaustive event because no other outcome is possible.

2. Trial and Event: Performing a random experiment is called a trial and outcome is termed an event. Tossing of a coin is a trial and the turning up of the head or tail is an event.

3. Equally likely events: Two events are said to be ‘equally likely’, if one of them cannot be expected in preference to the other. For instance, if we draw a card from a well-shuffled pack, we may get any card. Then the 52 different cases are equally likely.

4. Independent events: Two events may be independent, when the actual happening of one does not influence in any way the probability of the happening of the other.

5. Mutually Exclusive events: Two events are known as mutually exclusive, when the occurrence of one of them excludes the occurrence of the other. For example, on tossing a coin, either we get head or tail, but not both.

6. Compound Event: When two or more events occur in composition with each other, the simultaneous occurrence is called a compound event. When a die is thrown, getting a 5 or 6 is a compound event.

7. Favorable Events: The events, which ensure the required happening, are said to be favorable events. For example, in throwing a die, to have the even numbers, 2, 4 and 6 are favorable cases.

Odds in favor of an event and odds against an event-

If the number of favorable cases are ‘m’ and the number or not favorable cases are ‘n’.

Then-

1. Odds in favor of the event = m/n

2. Odds against the event = n/m

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