Probability, as the name suggests, deals with uncertainty. Probability of an event is defined as a positive number which denotes the chances of happening of an event. If an event is certain, i.e. there are 100% chances of that event to occur, then the probability of such event will be **one**. On the other hand, for an impossible event, i.e. there are no chances of that event to occur, the probability will be **zero**. Thus, we may summarize as:**1)** Probability of occurrence of an event is a number lying between 0 and 1.**2)** Probability of a certain event = 1.**3)** Probability of an impossible event = 0

4)

There are several terms and concepts related to this vast topic. In this page, we shall go ahead and learn about the probability and its applications in detail.

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- Complimentary events
- Conditional probability
- Compound events
- Independent and Dependent events
- Mutually exclusive events

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Probability in mathematics is the calculation of uncertainty. Probabilities are not at a single point. It is a number expressing the chances for a specific event will occur. Probability table is used to tabulate all the data calculated for a standard values from that values new probabilities can be determined. The events can also be tabulated. Learn more on facts of the concept with Tutorvista now.

Basic terms of probability are as follows:

**Probability: **Probability is the measure of how likely the events are occur.

**1) Experiment**: Experiment is a process by which an outcome is obtained

For Example: Tossing a coin

For Example: The sample space for a coin toss is, $S$ = $ {H,\ T}$

$\Rightarrow\ P(A)$ = $\frac{\text{Number of outcomes favorable}}{\text{Total numbers of outcomes}}$

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

for P(A $\cap$ B) $\neq$ 0

For independent events

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

For dependent events

P(A $\cap$ B) = P(A) $\times$ P(B|A)

4) Bayes Theorem

$P\left ( \frac{E_{i}}{A} \right )=\frac{P(E_{i})P\left ( \frac{A}{E_{i}} \right )}{\sum_{i=1}^{n}P(E_{i})P\left ( \frac{A}{E_{i}} \right )}$

where $E_i$ are n-mutually exclusive events with $P(E_i)$ ≠ 0

$P(x$ = $k)$ = $C(n,\ k)p^k\ (1\ -\ p)^{n\ -\ k}$

where $C(n,\ k)$ = $\frac{n!}{k!(n - k)!}$

Different rules are followed to find the probability of an event. These are also called as law of probability.

Rule 1: Rule of Subtraction

The probability that event A will occur is equal to 1 minus the probability that event A will not occur.

$\Rightarrow$ P(A) = 1 - P(A')

The probability the events A and B both occur is equal to the probability that event A occurs times the probability that Event B occurs, given that A has occurred.

$\Rightarrow$ P(A $\cap$ B) = P(A) P(B|A)

The probability the event A or event B occurs is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.

$\Rightarrow$ P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)

The probability simulations learning utilize the concepts of theoretical and experimental probabilities to solve problems involving uncertainty. The probability simulations learning to perform the model of real-world situations involving uncertainty. We can conduct probability simulations of the results for many problems of objects we pick should have the same number of results as the number of possible results of the problem, and all outcomes should be equally likely.

Theoretical probabilities of operations are performed in mathematically and describe what should happen. Experimental probability is calculated using data from tests or experiments. Experimental probability is the ratio of the number of times an outcome happened to the total number of events or trials. This type of ratio is labeled as the relative frequency.

If all the outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by

Experimental probability = $\frac{(Frequency\ of\ an\ outcome)}{(Total\ number\ of\ trials)}$ There are four types of probability are marginal probability, conditional probability, joint probability and union probability. The general law of multiplication used to compute joint probability and The general law of addition is used to compute the probability of union. The conditional law is used to compute conditional probability.

A probability tree is a diagram indicating probabilities and some conditional probabilities for combination of two or more events. Probability tree is one of the simple and easiest ways to solve a probability problems.

Venn diagrams are diagrams that show all possible logical relations between a finite collection. To draw a Venn diagram, first we draw a rectangle which is called your "universe".

$\Rightarrow$ P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)

This can be shown on a Venn diagram. The rectangle represents the sample space or universe, which is all of the possible outcomes of the experiment. The circle labelled as A represents event A. Similarly for B.

So, in the diagram, P(A) + P(B) is the whole of A (the whole circle) + the whole of B (so we have counted the middle bit twice). A' is the complement of A and means everything not in A. So P(A') is the probability that A does not occur.

Given below are some of the word problems on probability.

Total number of balls = 10

**Step 1:**

Since six out of ten balls are red

=> Probability that the first ball drawn is red = $\frac{ 6}{10}$.

**Step 2:**

If the first ball drawn is red, 9 balls are left and 5 out of them are red.

Similarly if the first two balls drawn are red colored, then the probability that the third ball drawn is red is $\frac{4}{8}$.

Hence by multiplication theorem, the required probability is

= $\frac{ 6}{10}$ * $\frac{4}{8}$

= $\frac{1}{6}$.

Since six out of ten balls are red

=> Probability that the first ball drawn is red = $\frac{ 6}{10}$.

If the first ball drawn is red, 9 balls are left and 5 out of them are red.

Similarly if the first two balls drawn are red colored, then the probability that the third ball drawn is red is $\frac{4}{8}$.

Hence by multiplication theorem, the required probability is

= $\frac{ 6}{10}$ * $\frac{4}{8}$

= $\frac{1}{6}$.

The event of James getting selected for the post is same as the event of John not getting appointed to the post. John not getting appointed to the post is the complementary event of John getting appointed.

The probability of John getting appointed as the Vice President = 0.47.

P(John not appointed) = 1 - 0.47 = 0.53.

Probability of James getting selected = 0.53.

The probability of John getting appointed as the Vice President = 0.47.

P(John not appointed) = 1 - 0.47 = 0.53.

Probability of James getting selected = 0.53.

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