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What is Probability?

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Probability is an important topic of mathematics that is widely used not only in mathematics but in several other fields. Probability of an event is the chances of that event to be happen. An event is supposed to be a process in the language of probability, such as - throwing a pair of dice, drawing a card from the deck of $52$, randomly selecting a marble from a jar of marbles of different colors etc.

Definition

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If there are n elementary events associated with an random experiment and m of them are favourable to an event E, then the probability of $E$ is denoted by $P(E)$ and is defined to the ratio $\frac{m}{n}$.

$\Rightarrow\ P(E)$ = $\frac{\text{Number of Favourable to Event A}}{\text{Total Number of Outcomes}}$

If an experiment has n equally likely outcomes in $S$ and $N$ of them are the event $A$, then the theoretical probability of event $A$ occurring is

$P(A)$ = $\frac{n}{N}$.

Probability Multiplication Rule

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If two events A and B from a sample S of a random experiment are mutually exclusive, then

$P(A\ \cup\ B)$ =$ P(A) + P(B)$

In this section, we examine whether such a rule exists, if $\cup$ is replaced by $\cap$ and $'+'$ is replaced by $'x'$ in the above addition rule. If it does exist, what are the particular conditions restricted on the events $A$ and $B$. This leads us to understand the dependency and independence of the events.
The Probability Multiplication Rule:
 
Consider events $A$ and $B$.

then $P(A\ \cap\ B)$ = $P(A)\ \times\ P(B)$.

Experimental Probability 

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The learning probability started with gambling, the probability mostly used in the field of physical science, commerce, biology science, medical science, weather forecasting, etc., an experimental having more than one possible outcome is called statistical experimental. Probability is a method which numerically measures the degree of uncertainty and, there, of certainty of the occurrences of an event. A statistical experimental is also known as a trial. In this article we shall learning an experimental probability.

Definition

Experimental probability of an event is the ratio of the number of times the event occurs to the total number of trials. The general meaning of the word probability is likelihood. If the event is absolutely certain the probability is said to be one or unity. If there is absolute impossibility of an event happening then probability is said to be zero. There are two ways of arriving at an actual measure of the probability of an event in the real life. They are mathematical probability and empirical probability, both of them falling under the category experimental probability, since they are based on the events and its performance. 
Formula:

$Experimental\ Probability$ = $\frac{\text{Number of times the event happen}}{\text{Total number of trials}}$

Probability Expected value

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Expected values are calculated as the weighted average of all possible outcomes using the probability of the outcomes as weights. The mean of a random variable is more commonly referred to as its expected value. The value obtain from some experiment whose outcomes are represented by the random variable. The expected value of a random variable $X$ is denoted by $E(x)$.
The random variable $X$ is discrete and has a probability distribution $f(x)$, the expected value of the random variable is given by:

$E(x)$ = $\sum\ x\ f(x)$.

Examples

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Example 1:

A coin is tossed $50$ times. $25$ times head appear. Find the experimental probability of getting heads.

Solution:

Step 1: $Experimental\ probability$ = $\frac{\text{Number of Times the Event Happen}}{\text{Total Number of Trials}}$

Step 2: Number of times heads appeared = $25$.

Step 3: Total number of experiments = $50$.

Step 4: Therefore experimental probability of getting a head = $\frac{25}{50}$ = $\frac{1}{2}$
Example 2:

John plays game his college. He did not score in 29 out of 58 games he played. What is the experimental probability of john score in a game?

Solution:

Step 1: Number of game john make score = $58 – 29$ = $29$

Step 2: $P\ (\text{John make a score})$ = $\frac{\text{Number of games in which John make goals}}{\text{Total number of games John play}}$

Step 3: = $\frac{29}{58}$

            = $\frac{1}{2}$

Step 4: Probability of john make a goal in a game is $\frac{1}{2}$.
Example 3:

A boy has four balls ($2$ black and $2$ red balls). What is the most likely mix of black and red ball.

What objects can be used to simulate the possible outcomes of the balls?

Find the theoretical probability that there are two red and two black balls.

Solution:

Assume that $P$(black ball) = $P$(red ball) = $\frac{1}{2}$

Objects can be used to simulate the possible outcomes of the balls?

Each ball can be black or red, so there are $2 \times 2 \times 2 \times 2$ or $16$ possible outcomes. 

Use a simulation that also has $2$ outcomes for each of $4$ events. One possible simulation would be to toss four coins, one for each ball, with heads representing black balls and tails representing red balls.

The theoretical probability that there are two red and two black balls.

There are $16$ probable outcomes, and the number of combinations that have two red and two black ball is $4C2$ or $6$.

So, the theoretical probability is $\frac{6}{16}$ or $\frac{3}{8}$