Probability

Experiment:Any activity which results in outcomes is called an experiment.

E.g. tossing a coin,rolling a die.

Event: An outcome of a random experiment is called an event.

FavorableEvent: The favorableoutcome is called favorable event.

e.g.Getting a head, getting even number on die.

Probability of an event:

* $\text{p}\left(\text{E}\right)\text{ = }\frac{\text{number of outcomes of favourable to E}}{\text{Number of all possible outcoes of the experiment}}$

If there are n elementary events associated with a random experiment and m of them are favorable to an event $\text{A}$, then the probability of happening or occurrence of event A is denoted by $\text{P}\left(\text{A}\right)$ and is defined as the ratio $\frac{\text{m}}{\text{n}}$.

Thus, $\text{P}\left(\text{A}\right)\text{ = }\frac{\text{m}}{\text{n}}$

Elementary event:

An event having only one outcome is called elementary event.

For e.g .getting a head

Complimentary event:

For an event $\text{E}$, the event “not” is called complimentary event.

It is denoted by $\overline{\text{E}}$.

$\text{P(}\overline{\text{E}}\text{) = 1 - P}\left(\text{E}\right)$

e.g.for event “getting even numbers”, the complimentary event is “ getting odd

numbers”

proof: If total no. of outcomes $\text{= n}$

total no. of favorable outcome of $\text{A = m}$

Total no.of non favorable $\text{= n - m}$

Then $\text{P}\left(\text{A}\right)\text{ = }\frac{\text{m}}{\text{n}}$

And $\text{P(}\overline{\text{A}}\text{) = }\frac{\text{n - m}}{\text{n}}$

$\to$ $\text{P(}\overline{\text{A}}\text{) = 1-}\frac{\text{m}}{\text{n}}$

$\to$ $\text{P}\left(\overline{\text{A}}\right)\text{ = 1 - P}\left(\text{A}\right)$

$\to$ $\text{P}\left(\overline{\text{A}}\right)\text{ + P}\left(\text{A}\right)\text{ = 1}$

Sure event:

The event which is sure or which has probability $\text{1}$ is called sure event.

Impossible event:

The event which is not possible or which has probability $\text{0}$ is called impossible event.

Range of probability

$\text{0}\le \text{P}\left(\text{E}\right)\le \text{1}$

As $\text{P}\left(\text{A}\right)\text{ = }\frac{\text{m}}{\text{n}}$

Clearly, $\text{0}\le \text{m < n}$.

$\text{0}\le \frac{\text{m}}{\text{n}}\text{ < 1}$

$\to$ $\text{0}\le \text{P}\left(\text{A}\right)\le \text{1}$

Sum of probabilities of all elementary:

Sum of probabilities of all elementary events is $\text{1}$.