Probability is the likeliness of an event to occur. It is the branch of mathematics that is studied by beginner and advanced mathematics students.
The probability of an event is determined by using two values 0 and 1. 0 means the probability of an event occurring is nill, so the event won’t take place. On the other hand, 1 means that the probability of an event occurring is 100%, so the event will definitely take place.
Consider an example. For instance, you toss a coin in the air. There are two possible outcomes either you’ll get a heads or tails. But how will you calculate the probability?
There’s a 50% chance that you get heads. So, the probability for this likeliness is ½.
When we talk about probability, we place the possibility of an event at 0 and 1. This is for certain events, just like the example of a coin above. However, not every event in life can happen with certainty and have two black-and-white outcomes.
Most events are uncertain and happen with a degree of randomness. For example, the probability of a baby being born on a leap year. For events like these, the probability falls somewhere between 0 to 1.
The probability of a baby being born on a leap year is 0.00065.
Before working with probability, you need to be familiar with the following essential terms.
A technique that you can repeat again and for which you cannot be absolutely certain of the result is called a random experiment. The settings and range of possible results should be the same for every repetition of the experiment and every repetition of a random experiment is referred to as a trial.
Example:
Coin flips and die rolls are two common instances of random experiments. Both rolling a die and tossing a coin are procedurally repeatable. Every time you roll a die, one of six unclear possible outcomes could appear: you could roll a 1, 2, 3, 4, 5, or 6. Every time you flip a coin, there are two possible results: heads or tails. These outcomes are unpredictable.
The outcomes of an experiment are determined at random. You are only able to view one outcome at a time because outcomes are mutually exclusive.
For instance,
When rolling a die, there are six possible outcomes: 1, 2, 3, 4, 5, and 6. Because you cannot roll a 1 and a 6 at the same time, these results are mutually exclusive. You only see one of the six possible outcomes for each roll.
Mutually exclusive events — Mutually exclusive events cannot occur at the same time. When one occurs, the probability of the other diminishes. Like, the weather can be hot or cold. These two outcomes cannot exist simultaneously.
The set of all potential results is known as the sample space. Typically, we use a Venn diagram or set notation (curly brackets {}) to represent a sample space.
As we can say,
When one die is rolled, the sample space is {1, 2, 3, 4, 5, 6}. The sample space for making a free throw is {make it miss it} in a similar manner.
The events in the sample space for which you all wish to determine a probability are known as favourable outcomes.
Example:
There is just one possible result that is beneficial, and that is 2; thus, you can compute the likelihood of rolling a 2 when you roll the dice. There are three positive outcomes that you might consider when calculating the likelihood of rolling an even number: 2, 4, and 6. There are five positive results that you can use to calculate the likelihood of rolling a number fewer than six: 1, 2, 3, 4, and 5.
Equally likely events — Equally likely events are those events whose probabilities of occurrence are equal. For instance, in tossing a coin, the probability of getting a head or a tail is equal. So, these are equally likely events.
A subset of the sample space to which a probability may be assigned is called an event. Similar to sample spaces, we frequently use a Venn diagram or set notation, such as curly brackets {}, to indicate events.
As an example
In a die-rolling experiment, there are various events that can occur: "rolling a 4," "rolling an even number," "rolling an odd number," and so forth.
When you talk about pulling a card from a deck of playing cards, the following events take place:
Exhaustive events — Exhaustive events are those in which all sets of results are the same as the sample space.
In mathematics, the most straightforward probabilities are those derived from experiments with several diverse but equally likely outcomes.
It is simple to determine the probability of events in such circumstances. All you have to do is divide the total number of possible outcomes by the number of favourable outcomes.
Probability of an Event = Number of Favorable OutcomesTotal Number of Possible Outcomes
As it turns out, this formula is very helpful for computing probabilities, as many probabilistic situations have equally likely outcomes.
There are some random experiments that have equally likely results, such as flipping coins, rolling dice, and drawing cards. This implies that the probability formula can be used.
Here are some practice questions and examples showing how to use the probability equation for coin tosses, dice rolls, and card draws.
Since there are two equally likely outcomes when tossing a coin, the sample space, or set of all potential outcomes, is as follows:
S= {Heads, Tails}
The likelihood that the event will "get tails" can be found using the probability formula. There is one positive outcome linked to this occurrence out of a total of two possible outcomes.
P(T) = Chance that a coin will land on tails = Number of Favorable OutcomesTotal Number of Possible Outcomes = 0.5 or ½
The same holds true for calculating the likelihood of receiving heads.
P(H) = Chance that a coin will land on heads = Number of Favorable OutcomesTotal Number of Possible Outcomes = 0.5 or ½
Six equally likely outcomes can occur while rolling a die; hence the sample space is:
Let S = {1, 2, 3, 4, 5, 6}.
Let's calculate the likelihood of rolling a number that is less than three using the probability formula. Since there are only two numbers on a die that are less than three, the likelihood of rolling any of those two numbers equals the probability of rolling a number that is less than three.
P(1 or 2) = Chance of rolling a number that is smaller than three = Number of Favorable OutcomesTotal Number of Possible Outcomes = 0.33 or 2/6.
Let's now consider pulling one card out of a deck of playing cards. All conceivable outcomes are equally likely, and there is an equal chance that you will draw any card from the deck. Since there are 52 cards in the deck, the sample space can have 52 different possible outcomes.
What is the likelihood of drawing the two hearts given the sample space? There is only one positive conclusion because there are only one 2 hearts in the entire deck.
As we have already established, there are 52 possible outcomes in all.
P(2♥) = Number of Favorable OutcomesTotal Number of Possible Outcomes = 1/52 or 0.019.
What about the odds of drawing a two, no matter what suit? Since there are four 2s in this deck, there are four possible winning combinations.
P(2) = Total Positive Results = Number of Favorable OutcomesTotal Number of Possible Outcomes = 4/52 or 0.076.
How likely is it that you will draw a diamond? There are 13 diamond-containing cards in the deck, as you can see if you count the diamonds in the sample space.
P(♦) = Number of Favorable OutcomesTotal Number of Possible Outcomes = ¼ = 0.25
Drawing a Spade or a Diamond
What is the likelihood of drawing a spade or a diamond? The deck is made up of 13 spades and 13 diamonds. Thus, there are twenty-six possible positive outcomes for this occurrence.
P(♦ or ♠ ) = Number of Favorable OutcomesTotal Number of Possible Outcomes = 26/52 = ½ = 0.5.
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For solving complex problems, the following types of probability need to be considered before anything:
It is centred on the possibility of anything happening. Scientific probability is based on the logic of probability.
There is a 50% statistical likelihood that a coin will land on the head. The statistical chance of an event is the likelihood that it will happen. The phrase theoretical probability refers to the outcome that is obtained by dividing the entire number of outcomes by the number of desirable outcomes.
It is based on an experiment's findings. By dividing the total number of trials by the entire number of possible outcomes, one can calculate the experimental chance.
For instance, the experimental probability of heads is 6/10 or 3/5 if a coin is flipped ten times, and the result is recorded as heads six times.
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In axiomatic probability, a set of rules or axioms is constructed that are applicable to all forms. These three axioms were developed by Kolmogorov and are referred to as his axioms.
Probability is measured using the axiomatic technique, which counts the chances of occurrences happening or not.
The concept of probability is used in Physics and Maths alike. In Dr. Feynman’s lectures, the probability theory was described as chances for an event to occur with the example of coin tossing as described above. It has its applications in the areas of Quantum Mechanics and Statistical Analysis.
Probability in Wave Function: In quantum mechanics, the wave function represents the state of a system. The square of the magnitude of the wave function gives the probability density function, which describes the likelihood of finding a particle at a certain position.
Heisenberg Uncertainty Principle: This principle states that there is a fundamental limit to the precision with which certain pairs of properties of a particle, such as position and momentum, can be known simultaneously. Probability distributions play a crucial role in understanding and interpreting this uncertainty.
Quantum Tunneling: Probability plays a key role in phenomena such as quantum tunneling, where particles can pass through classically forbidden energy barriers. Tunneling probabilities are calculated using quantum mechanical principles and play an important role in various technological applications.
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Microscopic vs. Macroscopic States: Statistical mechanics demonstrates the behavior between the microscopic properties of individual particles and the macroscopic properties of materials. Probability distributions are used to describe the distribution of particles among different energy levels and states.
Boltzmann Distribution: The Boltzmann distribution gives the probability that a system will be in a particular microscopic state at a given temperature. It forms the basis for understanding the thermodynamic properties of gases, liquids, and solids.
Entropy and Probability: Entropy, a measure of the disorder or randomness of a system, is closely related to the probability of different microstates. The second law of thermodynamics, which states that the entropy of an isolated system tends to increase over time, can be understood in terms of probabilities and statistical mechanics.
Probability can be calculated using the following formula:
Probability of an even t= Number of Favorable OutcomesTotal Number of Possible Outcomes
There are following basic rules of probability.
Addition rule: The probability of one or more events to occur.
mutually exclusive events: P(A or B) = P(A) + P(B)
not mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B)
Multiplication rule: The probability of all the events to occur.
independent events: P(A and B) = P(A) * P(B)
P(A and B) = P(A) * P(B|A)
Complement rule: The probability of anything besides one event.
For example, the probability of A not occurring 1-P(A)
Conditional probability: The probability of an event occurring with a condition that another event has already happened.
P(A|B) = P(A and B) / P(B)
A subfield of mathematics known as probability theory studies the likelihood of random events. Multiple outcomes are possible for random events to occur. It uses specific mathematical concepts to express the possibility of a given result.
Probability theory uses a few basic concepts to determine the chance that an event will occur, including sample space, probability distributions, random variables, etc.
The three probability theories are:
The probability of an outcome is never negative. It can be either 0 or 1 or something in between.
Probability concepts are easy to understand for those having a basic understanding of statistical concepts. To sharpen your mathematical skills and better understand probability, you can hire an online tutor. An online tutor in Statistics or Mathematics can greatly assist you in this regard.
In simple terms, probability is like a tool that helps us understand how likely things are to happen. Whether it's predicting if it'll rain tomorrow or figuring out the chances of winning a game, probability helps us make sense of uncertain situations.
We can better understand the world and make smarter decisions by learning about probability. So, whether you're flipping a coin or studying advanced physics, studying probability can be really helpful in everyday life. For this purpose, hire a certified mathematics tutor to learn more about it!
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