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# How to find probability for a discrete sample space?

The probability of a set of numbers is nothing but the measure of the possibility of occurrence of an event. In other words, it describes the certainty or uncertainty of an event. For this reason, the value of probability always lies between 0 and 1 where 0 means that the event is impossible while 1 means that there is hundred percent chance that the event will occur. The greater the probability of a number, the more chances it has of occurring. Also, the probability of a number can also never be a negative value because there is no such thing as a ‘negative probability’. Learning how to find probability for a given sample data will only be easy for students if they actually know how to define probability. A simple example of probability can be the tossing of the coin and the possibility of getting a head or a tail for each trial. Now here, both heads and tails have an equal possibility of showing up and therefore, the chance of occurrence of each of the two outcomes is 0.5 or half. Such an event is called an equally likely event where both the outcomes have an equal chance of coming true.

Knowing how to find probability is very important in today’s day and age. It is best used in risk analysis or different areas of finance where we need to study the financial market and invest in it by determining the probability of the risk factors involved. Probability theory is also employed in the field of game theory, machine learning, artificial intelligence, computer science and most importantly that of mathematics. In fact, quantum mechanics used the probability theory to define various phenomena occurring at atomic scales which proved to be a breakthrough of the twentieth century. It is also very commonly used in gambling where we are keen to know the chances of winning, losing or that of a tie. Of late, probability has also been used to analyse biological trends like those of the occurrence of diseases and to determine the spread of a disease. In the field of economics, probability helps in developing rigorous models on future growth and expansion. In other words, probability is everywhere around us and thus, it is important to know and determine how to find probability for a given statistical data. But, at first let us define the terms associated with probability in order to have a better understanding of the topic.

The first step in finding the probability of an event is to find out its sample space. Now, the sample space is described as the list of all the possible outcomes of that event. It means that we should be aware of the different possible outcomes that can occur for an event. For example, the sample space for the probability of finding the chance of occurring an even number on an unbiased dice will be all the numbers of the dice, that is, 1, 2,3,4,5 and 6. Similarly, in probability we often use the term mutually exclusive to describe an event. A mutually exclusive event is that one where no two events can occur together or at the same time. Like for example, flipping a coin is a mutually exclusive event because we cannot get a head and a tail at the same time for the same coin. There is another event known as an independent event which is used in probability. Such an event is the one where the occurrence of any outcome does not affect the chances of occurrence of the other ones. That is when a coin is flipped twice and a tail shows up, the chances of getting a head or tail in the next turn is not affected by the first tail. Knowing these terms is extremely important in order to learn how to find probability and to use it to solve mathematical and statistical problems.

The basic formula of probability is extremely simple and hardly requires any calculations. It can be calculated by finding the number of favourable outcomes of an event Y and dividing it by the total number of possible outcomes of the event Y. It is denoted by the symbol P(Y).

That is, P(Y) = favourable outcomes of Y/ Total outcomes in the sample space of Y.

Suppose we are asked to find the probability of the occurrence of the number 1 from the throw of a dice. Now, first we will find the sample space of the event of throwing a dice which is {1, 2, 3, 4, 5, 6}. These are total 6 in number while 1 occurs only once in this. Hence, the probability will be 1/6 or 0.16. The complement or the opposite of this event will be 1-(1/6) which comes out to be 5/6 and is denoted by the symbol ~Y. Anyone can easily learn how to find probability if they know how to determine the sample space for the data.

## How to find probability of independent events?

As described above, independent events are those which do not affect the occurrence of any of the other outcomes. Their probability can be easily found out by finding the intersection of the two set of events X and Y. Thus, P(X and Y) = P(X) X P(Y). This means that in an event where a dice is thrown twice, the probability of the occurrence of 1 will be (I/6) X (1/6) = 1/36. This is the most commonly used method that can be used to find the probability of many complicated problems in the field of mathematics and statistics.

## How to find probability of mutually exclusive events?

Similarly, we can also learn how to find probability for a set of mutually exclusive pair of events. However here, instead of the intersection of sets we make use of the union of sets to find probability. Its formula is given by P(X or Y) = P (XUY) = P(X) + P(Y). We can understand this with a simple example where we are asked to find the chances of getting a 5 or a 6 on throwing an unbiased dice. For such questions, we use the concept of mutually exclusive events to find the probability which is P (5 or 6) = P (5) + P (6) which is the sum of 1/6 + 1/6 and is equal to 2/6 or 1/3. This method is most commonly used to find the probability of drawing the cards from a deck or when the numbers are huge because of its extremely simple calculations involving only the addition of results.

Apart from this, we can also learn how to find probability of a given sample space using the method of conditional probability. The term conditional probability can be defined as the probability of occurrence of an event X given that the event Y has already occurred. It is denoted by the symbol P(X/Y) and also uses the intersection of the two sets to find probability. The basic formula of conditional probability is given by the expression: P(X/Y) = P (X ∩ Y)/ P(Y). Thus, the probability of Y cannot be zero or else the results would be undefined. This can be demonstrated using a simple example where we have to find the probability of passing the second test given that 42 people passed in the first test and 25 of them passed in both. Thus, P (second test/first test) = P (first test ∩ second test)/ (probability of first test) which is 25/42 and its conditional probability is 0.59.

### How to find probability using Bayes Theorem

Bayes Theorem is a very useful tool of probability that is applied in many mathematical questions. It can help us learn how to find probability for a given sample data by making use of the conditions related to that particular event. It is also denoted by the symbol P(X/Y) but uses a slightly different formula which is given by P(X/Y) = P(Y/X).P(X) / P(Y). Bayes Theorem may seem a bit complicated when applied for the first time, but can turn out to be extremely simple when explained through an example. Suppose we want to diagnose the test of a disease and it is found that 12% of the people have a lung disease which means that the P(X) = 0.12 while Y implies that 5% of the people visiting the centre are alcoholics, that is, P(Y) = 0.05. Also, we know that among the people with lung disease, about 7% of them are alcoholic which gives the value of P(Y/X) = 0.07 and now we are asked to determine the probability of the patient being an alcoholic if 7 percent of them have a lung disease. This is given by: P(X/Y) = 0.12 X 0.07/ 0.05 and the probability for this comes out to be 0.168.

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