Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from a deck, etc., then there must be some probability associated with that event. In terms of mathematics, probability refers to the ratio of wanted outcomes to the total number of possible outcomes. There are three approaches to the theory of probability, namely:
- Empirical Approach
- Classical Approach
- Axiomatic Approach
In this article, we are going to study about Axiomatic Approach. In this approach, we represent the probability in terms of sample space(S) and other terms.
- Random Event:- If the repetition of an experiment occurs several times under similar conditions if it does not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, then such an experiment is called a random event or a probabilistic event.
- Elementary Event – The elementary event refers to the outcome of each random event performed. Whenever the random event is performed, each associated outcome is known as an elementary event.
- Sample Space – Sample Space refers to the set of all possible outcomes of a random event.For example, when a coin is tossed, the possible outcomes are head and tail.
- Event – An event refers to the subset of the sample space associated with a random event.
- Occurrence of an Event – An event associated with a random event is said to occur if any one of the elementary events belonging to it is an outcome.
- Sure Event – An event associated with a random event is said to be sure event if it always occurs whenever the random event is performed.
- Impossible Event – An event associated with a random event is said to be impossible event if it never occurs whenever the random event is performed.
- Compound Event – An event associated with a random event is said to be compound event if it is the disjoint union of two or more elementary events.
- Mutually Exclusive Events – Two or more events associated with a random event are said to be mutually exclusive events if any one of the event occurs, it prevents the occurrence of all other events.This means that no two or more events can occur simultaneously at the same time.
- Exhaustive Events – Two or more events associated with a random event are said to be exhaustive events if their union is the sample space.
Probability of an Event – If there are total p possible outcomes associated with a random experiment and q of them are favourable outcomes to the event A, then the probability of event A is denoted by P(A) and is given by
P(A) = q/p
The probability of non-occurrence of event A, i.e, P(A’) = 1 – P(A)
- If the value of P(A) = 1, then event A is called a sure event .
- If the value of P(A) = 0, then event A is called an impossible event.
- Also, P(A) + P(A’) = 1
It says the possibility of an event happening is equal to the ratio of the number of favorable outcomes and the total number of outcomes.
Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes
- General – Let A, B, and C are the events associated with a random experiment, then
- P(A∪B) = P(A) + P(B) – P(A∩B)
- P(A∪B) = P(A) + P(B) if A and B are mutually exclusive
- P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C)- P(C∩A) + P(A∩B∩C)
- P(A∩B’) = P(A) – P(A∩B)
- P(A’∩B) = P(B) – P(A∩B)
- Extension of Multiplication Theorem – Let A1, A2, ….., An are n events associated with a random experiment, then P(A1∩A2∩A3 ….. An) = P(A1)P(A2/A1)P(A3/A2∩A1) ….. P(An/A1∩A2∩A3∩ ….. ∩An-1)
Probability Density Function
The Probability Density Function (PDF) is the probability function that is the density of a continuous random variable that lies between a range of certain values. Probability density function explains how the normal distribution and how mean and deviation exist in the system.
Problems and Solutions on Probability
Example-1: A bag contains 10 oranges and 20 apples out of which 5 apples and 3 oranges are defective. If a person takes out two at random, what is the probability that either both are good or both are apples?
Out of 30 items, two can be selected in 30C2 ways.
Thus, Total elementary events = 30C2 .
Consider the events: A = Getting two apples B = Getting two good items
Required Probability is : P(A∪B) = P(A) + P(B) – P(A∩B) …(I)
There are 20 apples, out of which 2 can be drawn in 20C2 ways.
P(A) = 20C2/30C2
There are 8 defective items and 22 are good, Out of 22 good items, two can be drawn in 22C2 ways .
P(B) = 22C2/30C2
Since there are 15 items that are good apples, out of which 2 can be selected in 15C2 ways .
P(A∩B) = 15C2/30C2
Substituting the values of P(A), P(B) and P(A∩B) in (I)
Required probability is = (20C2/30C2) + (22C2/30C2) – (15C2/30C2) = 316/435
Example-2: The probability that a person will get an electric contract is 2/5 and the probability that he will not get a plumbing contract is 4/7. If the probability of getting at least one contact is 2/3, what is the probability of getting both?
Consider the two events:
A = Person gets electric contract
B = Person gets plumbing contract
P(A) = 2/5
P(B’) = 4/7
P(A∪B) = 2/3
P(A∩B) = P(A) + P(B) – P(A∪B) = (2/5) + (1 – 4/7) – (2/3) = 17/105
Total Law of Probability – Let S be the sample space associated with a random experiment and E1, E2, …, En be n mutually exclusive and exhaustive events associated with the random experiment. If A is any event that occurs with E1 or E2 or … or En, then
P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + ... + P(En)P(A/En)
Example-1: A bag contains 3 black balls and 4 red balls. A second bag contains 4 black balls and 2 red balls. One bag is selected at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is red.
A red ball can be drawn in two ways:
- Selecting bag I and then drawing a red ball from it.
- Selecting bag II and then drawing a red ball from it.
Let E1, E2 and A be the defined events as follows :
E1 = Selecting bag I E2 = Selecting bag II A = Drawing red ball Since selecting one of the two bags at random .
P(E1) = 1/2 P(E2) = 1/2
Now, probability of drawing red ball when first bag has been chosen P(A/E1) = 4/7 and,
probability of drawing red ball when second bag has been chosen P(A/E2) = 2/6
Using total law of probability, we have
P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) = (1/2)(4/7) + (1/2)(2/6) = 19/42
Hence, the probability of drawing a red ball is 19/42
Example-2: In a bulb factory, three machines namely A, B, C produces 25%, 35% and 40% of the total bulbs respectively. Of their output, 5, 4 and 2 percent are defective bulbs respectively. A bulb is drawn at random from products. What is the probability that bulb drawn is defective?
Let E1, E2, E3
and A be the defined events as follows :
E1 = The bulb is manufactured by machine A
E2 = The bulb is manufactured by machine B
E3 = The bulb is manufactured by machine C
A = The bulb is defective According to given conditions ; P(E1) = 25/100 P(E2) = 35/100 P(E3) = 40/100
Now, probability that the bulb is defective given that is produced by Machine A
P(A/E1) = 5/100
and, probability that the bulb is defective given that is produced by Machine B
P(A/E2) = 4/100
and, probability that the bulb is defective given that is produced by Machine C
P(A/E3) = 2/100
Using total law of probability, we have
P(A) = P(E1)P(A/E1) + P(E2)P(A/E2) + P(E3)P(A/E3) = (25/100)(5/100) + (35/100)(4/100) + (40/100)(2/100) = 0.0345
Hence, the probability that the bulb is defective is 0.0345
Uses of Probability
Probability is important and refers to the extent of occurrence of events. Prediction of future events and actions can be known accordingly. Here are some of the uses of probability in our day-to-day life.
- Weather Forecasting– Whether it is cloudy, sunny, stormy, or rainy, on the basis of the prediction made, we plan our day. If the weather forecast says that there is a 75% chance of rain today. With this, a question arises how is the calculation of probability or precise prediction done? Another example, if in a company 100 employees are working for the IT department and 100 are working for the content writer and 30 are absent, then how many are absent from the IT department?
- Agriculture- The occurrence of erratic weather is beyond human control, but it is possible to prepare for adverse weather if it is forecasted beforehand. Farmers sow seeds on the basis of the assumption that in which month it will rain and how much, here also probability is used.
- Politics– Every Politician wants to predict the outcome of an election even before the polling is done or before the counting of the votes. Sometimes they predict which political party will rise to power by closely studying the results of exit polls. There are other good uses of probability, like predicting the number of students who would be needing jobs in the upcoming year so that the vacancy can be created accordingly. Politicians can also analyze the rate of car and bike accidents increased in past years so that they can take measures and reduce road accidents.
- Insurance– Insurance companies also use probability to find out the chances of a person’s death or also insurance for a vehicle given is also predicted by studying the database of the person’s family history and personal habits like drinking and smoking.
FAQs on Probability
Q1. How to solve Probability?
To solve probability, frist of all identify an event having one result. Then find all the possible results present, step 3 is to divide the number of favourable outcomes by the total number of possible outcomes.
Q2. What do you mean by probability ?
How likely something is to happen is probability. Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are.
Q3. What are the 3 types of probability?
There are 3 major probabilities
- Experimental Probability.
- Axiomatic Probability.
- Theoretical Probability.