Probability in Maths: The Ultimate Guide

'Uncertainty' is a word that fills us all with dread, and often, we are bound to make day-to-day decisions based on mere chance. But what if there's a way to narrow down our chances into the realm of certainty? What if you could bridge the gap between uncertainty and possibility with confidence? The mathematical field of Probability helps you achieve that with great precision. 

From everyday scenarios such as weather predictions to more complex applications in science, politics and finance, Probability offers a precise glimpse of the most likely outcomes. This blog dissects this concept in detail covering its key terminologies, formulas, theorems and more. So read on, take a chance at mastering the art and science of Probability, and become a pro at prediction! 

Table of Contents

1) What is Probability? 

2) Key Terms in Probability 

3) Probability of an Event 

4) Probability Formulas 

5) Different Types of Probability 

6) Probability Theorems 

7) Example of Probability 

8) Applications of Probability 

9) Conclusion 

What is Probability? 

Probability defines the likelihood of an event's occurrence. There are countless real-life situations which may compel us to predict the outcome of an event, but we may not be sure of the result. In such cases, this event is said to be probable to occur or not occur. Probability boasts great applications in business and games to make predictions, with extensive applications in this field of Artificial Intelligence (AI).
 

Key Terms in Probability 

Here are some basic concepts or terminologies used in Probability: 

1) Sample Space: This is the set of all possible outcomes in a Probability experiment. For instance, in the case of a coin toss, it’s “head” and “tail”. 

2) Sample Point: This defines one of the possible results of an experiment. For example, sample points are 1 to 6 when rolling a fair six-sided dice. 

3) Experiment: This refers to a process with uncertain results. Examples include card selection, coin tossing, or rolling a die. 

4) Event: This defines a subset of the sample space representing specific outcomes, such as getting “3” when rolling a die. 

5) Favourable Outcome: This outcome produces the desired or expected consequence. 

Probability of an Event 

An event in Probability theory is a set of outcomes of an experiment (or a subset of the sample space). If P(E) represents the Probability of an event E, then: 

1) P(E) = 0 if and only if E is an impossible event. 

2) P(E) = 1 if and only if E is a certain event. 

3) 0 ≤ P(E) ≤ 1 

Suppose there are two events named "A" and "B." The Probability of event "A" is higher than "B" (P(A) > P(B)) if and only if event "A" is likelier to occur than event "B." 

Probability Formulas 

The Probability formulas for the events A and B are summarised below: 

Different Types of Probability 

There are three major types of probabilities, namely theoretical Probability, experimental Probability and axiomatic Probability. These are explored in detail below 

Theoretical Probability 

Theoretical Probability is based on the possible chances of something happening. It's mainly based on the reasoning behind Probability. For instance, if a coin is tossed, the theoretical Probability of getting a head will be 

1212

Experimental Probability 

Experimental Probability can be determined by dividing the total number of trials by the number of potential outcomes. For example, if a coin is flipped 12 times and heads are reported four times, the experimental chance of heads is 

412412

or

1313

Axiomatic Probability 

In axiomatic Probability, a set of rules or axioms are applied to all the types. The chances of occurrence or non-occurrence of any event is quantified by applying these axioms as explained below: 

a) The smallest possible Probability is zero, and the biggest is one. 

b) A certain event has a Probability equal to one. 

c) Any two mutually exclusive events can't occur simultaneously, while the union of events says only one can occur. 

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Probability Theorems 

Some of the most important Probability theorems are summarised below:
 

Bayes' Theorem on Conditional Probability  

Bayes’ theorem describes the Probability of occurrence of an event related to any condition. It is also considered for the case of conditional Probability. Bayes theorem is also the formula for the Probability of “causes”. 

For example, suppose you are calculating the likelihood of taking a green ball from the second bag out of three different bags of balls. Each bag contains three different-coloured balls (Green, blue, and black). In that case, the Probability of an event's occurrence is calculated based on other conditions, known as conditional Probability. 

Formula for Bayes' Theorem 

If A and B are two events, the formula for Baye’s theorem is as follows: 

P(A|B|) = P(B|A|)P(A)P(B) Where P(B)≠0PAB = PBAPAPB Where PB≠0

Here, P(A|B) is the Probability of condition when event A occurs while event B has already happened. 

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Example of Probability 

Here are a couple of examples to illustrate the concept of Probability 

Example 1: Let's say there are eight balls in a container, out of which four are red, three are blue and one is yellow. Then, the Probability of picking a red ball can be determined by dividing the number of red balls in the container by the total number of balls in the container. So, the answer is  

4848

or 

1212

Example 2: Let’s say you have to calculate the Probability that an even number is obtained when you roll a dice. You must consider the six possible outcomes when a fair six-sided dice is rolled: 1, 2, 3, 4, 5, or 6. Out of these possible outcomes, half are even (2, 4, 6) and half are odd (1, 3, 5). Therefore, the Probability of getting an even number is: 

1) P(even) = 

Number of even outcomesTotal number of outcomes / Number of even outcomesTotal number of outcomes

2) P(even) = 

3636

3) P(even) = 

1212
 

Applications of Probability 

Probability is utilised across various fields, from weather forecasting to politics, as outlined below: 

1) Weather Predictions: Meteorologists use Probability to predict the chance of weather conditions like snow or rain occurring in a particular area on a certain day. For example, a prediction could indicate a 75% likelihood of rain between 3 PM and 6 PM today, showing a high chance of precipitation during that period. 

2) Sports Betting: Probability is a useful tool for betting firms when determining the odds for upcoming games. If analysis indicates that one team is 90% likely to win and the other has a 10% chance, the betting company will offer bigger payouts for bets on the underdog team. 

3) Sales Forecasting: Probability helps sellers predict the likelihood of selling specific amounts of products within a set timeframe. For instance, a model could forecast an 80% chance of selling a minimum of 100 items on a specific day. These statistics help determine inventory choices. 

4) Health Insurance: Insurers can utilize a probabilistic model to estimate the likelihood of individuals experiencing certain annual medical expenses. People who are more likely to spend money will have to pay higher premiums because they are expected to be more costly to insure. 

5) Natural Disasters: Evaluation of natural disasters, such as tornadoes or hurricanes, is conducted by environmental agencies. Strong chances of such disasters guide the decisions on resource allocation and housing measures to minimise potential effects. 

6)  Investing: Investors use Probability to assess the likelihood of earning returns from investments. For instance, a potential investor could evaluate a 2% likelihood of Company A's stock rising by 50 times within the next year. 

7) Politics: Analysts in politics use probabilities to assess the likelihood of various candidates winning elections. An estimation could show that Candidate A has a 50% Probability of winning, Candidate B has a 10% Probability, and Candidate C has an 8% Probability. This assists voters in evaluating the Probability of success for each candidate. 

Conclusion 

In conclusion, mastering Probability is a powerful way to understand the uncertainties we experience on a daily basis. More than just numbers, it's about embracing the unpredictable facets of life. Whether you’re rolling a dice or calculating risks for your business, this blog's exploration of Probability will guide you to handle uncertainties with precision. 

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