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The probability of a given event is an expression of the possibility of occurrence of an event. A probability is a number that ranges from 0 to 1. Zero for an event that cannot occur and 1 for an event that 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 understanding the meaning of probability, and today, it has become one of the essential tools of statistics.
Question 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
Let us understand probability with the dice examples.
Question 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.
When the event E has only one sample point of a sample space. Then it is called a simple event. Besides, this is the event that 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}.
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 number on the dice), as there is more than one possibility, there are three possibilities i.e., E = {2,4,6}
It is the probability where we are sure about the experiment to occur in a certain event. A certain event always holds the probability of 1.
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.
In this kind of event, 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.
In this kind of event, 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.
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 )
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 SamplingIn this, we split the subjects into mutually exclusive groups. After that, we use simple random sampling to choose members from the groups.
Systematic SamplingIn 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 SamplingIn 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 selects the population based on the selected region.
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.
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.
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.
The normal distribution is used in the natural and social sciences to represent real-valued random variables whose distributions are not known.
The Poisson distribution can be used to calculate the probabilities of various numbers of "successes" based on the mean number of successes.
It is also called the negative exponential distribution. It is a probability distribution that describes the 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.
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- the weight of a person. The weight can be any value from 5o kgs, or 52.5 kgs, or 52.5436kgs.
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 great at writing probability assignments.
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Arithmetic Mean |
Inequalities and convergence |
Distributions of random variables |
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Characteristic functions |
Bayes Theorem |
Metric Spaces |
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