Introduction to Probability with Examples and Solutions

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Probability is a section of math that is essential in various aspects of life. Here we discuss the concept of probability with examples and solutions.

General Concept on Probability

In our day to day life we use probability though we all are not familiar with that term or the part of mathematics. We pronounce it to consider whether any incident will happen certainly or not. So now we will discuss the basic concepts which will definitely help anyone understand the probability.


Introduction to the Concept of Probability with Examples and Solutions
Probability

What is probability?

Probability is just a number which indicates the chance of any event. Any event can occur certainly or cannot happen. If any incident happens certainly, probability will be 1 and probability will be 0 for not happening certainly. Sometimes any incident can have both the chance of occurring and not occurring. In this case, probability shows the value between 0 & 1. For this reason, probability varies from 0 to 1. If we want to express it in terms of percentage, it ranges from 0% to 100%. Some outcomes from the sample space about which we are concerned with. These outcomes are called favorable outcomes. They are a portion of total outcomes. So, probability is defined by-    

Introduction to the Concept of Probability with Examples and Solutions

Terms Related to Probability

Sample Space

When we do any experiment, we get a set of possible outcomes. Sample space means this set of possible outcomes. It is generally represented by S. Each set has a number of elements. Each element is called a member of the sample space or simply a sample point. If you toss a coin, you will get two possible outcomes such as head and tail. This sample space will be S = {head, tail} Or, simply S = {H, T} When you throw any dies, you will get six possible outcomes and they are 1,2,3,4,5,6. Simple space would be written as S = {1,2,3,4,5,6} Where 1,3,5 are odd numbers and 2,4,6 are even numbers. So, we can write it as S = {odd, even}

Events

Event is a set of favorable outcomes of fixed characteristics in an experiment. It is a subset of sample space. Event can have all the value or some value or no value of sample space. F = {P: P is an even factor of 9} 9 has three factors which are 1,3,9. As there is no even factor of 9 , it will be definitely null set. F = {P : P is factor of 9 which is greater than 3} For the above case sample points are 1,3,9 and favorable points for the event is only 9. So, event is the subset of sample space. F = {P: P is the factor of 9} Event is comprised of all the values of sample space.

Complementary Events

Occurring of any event and not occurring of any event are mutually complementary events. In tossing of any coin the event of getting head is complementary with the event of getting tail.

Equally likely Events

In an experiment, if probability of occurring all events is equal is called equally likely events. One event’s probability is not more or less than other’s probability. Suppose you have thrown a dice. The probability of getting any of 1,2,3,4,5,6 on the upper side is equal and this is 1/6.

Mutually Exclusive Events

Two events will be mutually exclusive only then when there is no common points. If two events X & Y are mutually exclusive                                   X∩Y = P(X∩Y) = 0 In a coin tossing, the probability of getting head and tail is a mutually exclusive events.

Not Mutually Exclusive Events

Two events are such that they can be occurred simultaneously because of their some common sample points. These types of events are known as not mutually exclusive events. In an experiment of throwing dies if A = {1,2,3,4} and B = {3,4,5,6} these two events are not mutually exclusive.

Sure Events

In a random experiment, if any event is such that it occurs in every case of an experiment is called sure event. In a sure event, the number of favorable sample points is equal to the number of total sample points. If a coin is tossed, the probability of getting head or tail is always 1. Because you will get head or tail after tossing a coin. They are equally likely events and will happen surely.

Uncertain Events

If any event is such that it could happen in any case or could not happen in other case is called uncertain event. i.e. if any event does not occur in every case, it is uncertain event. The probability of happening this type of event is greater than 0 but less than 1. Let, A is an uncertain event so probability of happening A event P(A) is 0 < P(A) < 1 If you throw a dies, you will not get even numbers always. So, probability of getting even numbers is always an uncertain event.

Impossible Events

In a random experiment, if any event is such that this will not happen in any case of experiment, this type of imaginary event is known as impossible event. Impossible event has no favorable sample point so probability of impossible sample points is always 0. If you throw a dies, event of getting 8 is an impossible event.

Independent Events

We obtain some events from an experiment. When the probability of occurring any event does not depend on occurring or not occurring of other events, these type of events are independent events. The probability of occurring two events simultaneously is equal to the multiplication of occurring any event individually. If A & B are independent events, the probability of occurring A event is P(A) and probability of occurring B event is P(B). So, probability of occurring A and B event simultaneously P(A∩B) is defined by P(A∩B) = P(A) . P(B) Two students are asked to solve the math. So, probability of solving the math by one student does not depend on other one.

Dependent Events

Two events from an experiment are such that occurring of one event depends on occurring or not occurring of other event, these types of events are known as dependent events. Let, A & B are two events. If occurring of B event depends on occurring of A event, B event is called dependent event. Probability of occurring B event is defined by - 


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Know World Now: Introduction to Probability with Examples and Solutions
Introduction to Probability with Examples and Solutions
Probability is a section of math that is essential in various aspects of life. Here we discuss the concept of probability with examples and solutions.
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