Introduction to Probability

In this article, we will learn some of the basic terminologies of the probability like random experiment, outcome, sample space, trial & events, random variable and many more.

Probability is the science of decision-making with calculated risk in face of uncertainty. Probability is a measure of the likelihood that a given event will occur. It is quantified as a number between 0 and 1, inclusive – where 0 indicates that the event will not occur and 1 indicates that the event will occur.

In this article, we will learn about the basics of probability.

• Probability
• Terminology
  • Random Experiment
  • Outcome
  • Sample Space
  • Trial and Events
  • Random Variable
• Formula for Probability
• Difference between Probability and Likelihood
• Conditional Probability
• Application of Probability

Probability

Probability is simply “How likely is something to happen”.

Example: getting 6 on throwing a die.

Terminology

Random Experiment:

If in each trial of an experiment conducted under identical conditions, and the outcomes are not known, then such an experiment is known as a Random Experiment. 

Example: Tossing a coin, throwing a die

Outcome:

The result of a random experiment is called Outcome.

Example: In tossing a coin, there are only 2 possible outcomes Head (H) and Tail (T)

Sample Space:

The set of all possible outcomes of a random experiment is called Sample Space.

Example: Sample space of rolling dice: {1, 2, 3, 4, 5, 6}

 Trial and Events: 

• Any particular performance of a random experiment is called a trial.
• A combination of outcomes is termed as events.

Let’s understand Trail and Event by an example of dice:

• Rolling dice is a trial
  • Possible outcomes {1, 2, 3, 4, 5, 6}
• Possible events corresponding to rolling dice
  • Odd number of points: {1, 3, 5}
  • Even number of points: {2, 4, 6}
  • Points between 2 and 6: {3, 4, 5}

Types of Events:

• Exhaustive Events:

  Total number of distinct possible outcomes of a random experiment is known as the exhaustive events.

Example: In tossing coin, there are 2 exhaustive events head and tail.

• Favorable Events:

  Those outcomes of trials in which a given event may happen are called favorable cases for that events.

Example: Rolling two dice

Number of cases favorable to getting a sum of 7

(1,6), (2,5), (3,4), (4,3), (5,2), and (6,1)

• Mutually Exclusive Events:

  Events that do not occur at the same time are called Mutually Exclusive Events.

Example: Tossing a coin, the event Head and Tail are mutually exclusive events.

• Equally Likely Events:

  Events that have the same probability of occurrence are called Equally likely Events.

Example: All 6 outcomes on throwing a dice are mutually exclusive

• Independent Events:

  An event that doesn’t depend on another event in any random experiment is known as an independent event.

Example: In throwing a coin, getting Head and Tails both of these events are independent.

Random Variable:

A real-valued function defined over the sample space of a random experiment is called Random Variable.

A random variable is classified into 2 parts:

• Discrete Random Variable 
• Continuous Random Variable

To know more about random variables, read the article on Probability Distributions.

The Formula for Probability

Probability of an event (E) is,  ratio of the number of favorable outcomes and a total number of possible outcomes.

Let’s understand the formula by an example:

Problem : Tossing three coins

Find the probability of getting exactly two heads and one tail

Solution: Sample Space for tossing three coins:

S = { HHH, HHT, HTH, THH, THT, TTH, HTT, TTT}

So, the total number of possible outcomes = 8

E = Event of getting exactly two number of heads and one tail

Favorable outcomes = { HHT, HTH, THH}

Number of Favorable outcomes = 3

Difference Between Probability and Likelihood

Probability attach to possible results (chances).

Likelihood attaches to the hypothesis.

Let’s understand the difference by an example of cricket,

Problem: Captain have to decide to bat first

Probability: Only two possibilities 

• Choose to Bat
• Doesn’t choose to Bat 

• P(choose to bat) = P(doesn’t choose to bat) = ½ = 0.5

Likelihood: Choosing to bat first will depend on

• Weather Conditions ( Rainfall, wind speed)
• Due on Pitch
• Humidity

Conditional Probability:

Condition Probability is the probability of an event occurring given that another event has already occurred.

Example: Monte Hall Problem

Application of Probability

• Weather forecasting
• Stock Prediction
• Risk assessment in Catastrophic Modelling
• Predicting the chance of winning in various games

Conclusion

In this article we briefly mention about the basics of probability with examples. This article includes random experiment, random variable, events distribution and its application.

Hope you will like the article.

Keep Learning!!

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