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Probability Definition

The probability of a given event is an expression of the possibility of occurrence of an event. A probability is a number which ranges from 0 to 1. Zero for an event which cannot occur and 1 for an event which is assured to occur. The joint probability of events occurring can be calculated based on the type of events. These events are with respect to each other - Independent events, mutually exclusive events, conditional probability, not mutually exclusive events, inverse probability, and many more.

In other words, we say probability and statistics are so fundamentally interrelated with each other. It is challenging to discuss statistics without an understanding of the meaning of probability. Probability today has become one of the essential tools of statistics.

Probability Examples And Solutions

Q: Suppose that you have 10 marbles in a bag: 6 are blue, 2 are green, 1 is yellow, and 1 is red. Find out the probability of how many times the blue marble gets picked?

Solution: Number of ways it can happen: 8 (there are 8 blues) Total number of outcomes: 10 (there are 5 marbles in total)

So the probability = 8/10 = 0.8

Lets us understand probability with the dice probability examples.

Q: Suppose that you have rolled the two six-sided dice. Find out the probability that the sum of the two dice is seven?

Solution: The best and most effective way to solve this problem is to take the help of the table given above. Look at this in each row; there is one dice roll. Where you can see the sum of the two dice is equal to seven. Since we have the six rows, so there are six possible outcomes where we have the sum of two dice is equal to seven. It is the number of total possible outcomes remains 36. Now let’s find out the probability of seven. It will be 6/36; thus, it will be ⅙.

Types of Events

1. Simple Event

When the event E has only one sample point of a sample space. Then it is called simple event. Besides, this is the event which consists of precisely one outcome. For example: Suppose that you throw a die, the possibility of 5 appearing on the die is a simple event and it is given by E = {2}.

2. Compound Event

As the name suggests, it is having more than one sample point on a sample space. In other words, it involves combining two or more events thus it helps in finding the probability of such a combination of events.
For example, suppose that you throw a die, the possibility of an even number appearing is a compound event (because there are more than one even numbers on the dice), as there is more than one possibility, there are three possibilities i.e., E = {2,4,6}

3. Certain Event

It is the probability where we are sure about the experiment to occur is a certain event. The certain event always holds the probability of 1.

4. Impossible Event

This type of probability applies to the event which cannot occur. In other words, there is no chance of the event to occur; this is known as an impossible event. This type of event holds the probability of 0. Here is the best example to elaborate on the impossible event. When you draw two cards from the deck, then there will be an impossible event that both cards are red and black.

5. Equally likely Events

In this kind of events, the outcome of the events are equally likely to happen. For example when you toss a coin then there is always equally likely event to get heads or tails.

6. Complementary Events

In this kind of events, the outcome of the events is equally likely to happen. For example, when you toss a coin, then there is always an equally likely event to get heads or tails.

7. Mutually Exclusive Events

When we do have two events that can not occur at the same time is known as mutually exclusive events. These kinds of events always produce different outcomes.
If A and B are two events, then
( A ∩ B ) = Ø
P ( A ∩ B ) = 0
P ( A ∪ B) = P (A) + P ( B )

Types of Probability Sampling

Simple random sampling

It is a completely random method of selecting the subjects. In this, we assign numbers to all subjects. After this, we use a random number generator to choose random numbers.

Stratified Random Sampling

In this, we split the subjects into mutually exclusive groups. After that, we use simple random sampling to choose members from the groups.

Systematic Sampling

In this sampling, we choose every “nth” participant from a complete list. For example, you can select every 10th person listed on our list.

Cluster Random Sampling

In this, we randomly select participants from an extensive list. It is also too large for simple random sampling. For example, if you wanted to choose 500 participants from the entire population of a country. It is nearly impossible to have a complete list of everyone. For this, the researcher randomly selects areas and then randomly select the population based on the selected region.

Types of Probability Distributions

Bernoulli Distribution:

It is a discrete distribution having two possible outcomes considered by n=0 and n=1 in which n=1("success") occurs with probability p and n=0("failure") occurs with probability q=p-1, where 0

It therefore has probability density function.

Uniform Distribution:

A uniform distribution is also known as a rectangular distribution. It is a probability distribution that has constant probability where variable x has a uniform distribution, denoted U(a, b) if its probability density function is f(x)=1b−a.

Binomial Distribution:

It is a type of distribution that has two possible outcomes, i.e. (Success or Failure). For example, a coin toss has only two possible outcomes: heads or tails. Binomial formula b(x; n, P) = nCx * Px * (1 - P)n - x.

Normal Distribution:

Normal distribution is used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

Poisson Distribution:

The Poisson distribution can be used to calculate the probabilities of various numbers of "successes" based on the mean number of successes.

Exponential Distribution:

It also called the negative exponential distribution. It is a probability distribution that describes time between events in a poisson process. A continuous random variable X is said to have an Exponential (λ) distribution if it has probability density function fX(x|λ) = λe−λx for x > 0

Where λ > 0 is called the rate of the distribution.

Common Data Types

Discrete Data: It can take only specified values. For example, when you roll a die, the possible outcomes are 1, 2, 3, 4, 5 or 6 and not 1.5 or 2.45.

Continuous Data: It can take any value within a given range. The range may be finite or infinite. For example- weight of a person. The weight can be any value from 5o kgs, or 52.5 kgs, or 52.5436kgs.

What Are The Topics Covered By Our Experts?

We also covered probability related topics given below. You can seek probability homework help or probability assignment help on any concept of probability from our professional experts. They are the great at writing probability assignments.

            

Arithmetic Mean

Inequalities and convergence

Distributions of random variables

Characteristic functions

Bayes Theorem

Metric Spaces

            

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