Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. The main idea of probability is to measure the likelihood of an event occurring. By understanding probability, we can make predictions, assess risks, and obtain information about various events.

Let's begin our exploration of probability theory by understanding some fundamental concepts.

What is probability?

Probability measures the uncertainty of events. It provides a numerical measure, ranging from 0 to 1, that describes the likelihood of a certain event occurring. We use the concept of probability in many real-world situations, such as weather forecasting, gambling, insurance policies, and decision-making processes.

Here's a simple definition of probability:

P(event) = (number of favorable outcomes) / (total number of possible outcomes)

For example, consider throwing a six-sided die. The probability of a particular number, say 3, coming up is:

P(roll 3) = 1/6

Basic terminology

To better understand probability theory we need to be familiar with some basic terminology:

Use

An experiment is an action or process that results in one or more outcomes. Common examples are tossing a coin, rolling dice, or drawing a card from a deck.

Outcome

An outcome is the result of an experiment, such as getting heads when tossing a coin.

Sample space

The sample space, often denoted by S, is the set of all possible outcomes of an experiment. For example, in a coin toss, the sample space is S = {'Heads', 'Tails'}.

Events

An event is a subset of the sample space. It may include one or more outcomes. For example, getting an even number when throwing a dice is an event that includes the outcomes {2, 4, 6}.

Types of events

There are different types of events in probability:

Simple event

A simple event has only one outcome. Example: A 4 comes up on a die.

Mixed event

A compound event is a combination of two or more simple events. Example: Getting an even number and getting a number greater than 4.

Fixed event

An event that will definitely happen. Example: A number between 1 and 6 will come up on a dice.

Impossible event

An event that cannot happen. Example: Rolling up a 7 on a six-sided die.

Mutually exclusive events

Events that cannot happen at the same time. Example: Heads and tails on tossing the same coin.

Independent events

Two events are independent if the occurrence of one does not affect the probability of the other. Example: tossing a coin and throwing a dice.

Understanding probability with examples

Let us understand how probability works mathematically through some examples:

Example 1: Tossing a coin

You have a fair coin, and you want to find the probability of getting heads when you toss it.

Sample space: S = {'Heads', 'Tails'}

The probability of getting heads is calculated as follows:

P(heads) = Number of outcomes with heads / Total number of outcomes = 1/2

Example 2: Throwing a dice

Now, you have a six-sided die, and you want to find the probability of getting a number greater than 4.

Sample space: S = {1, 2, 3, 4, 5, 6}

When a number greater than 4 comes up, favorable outcomes are 5 and 6.

Number of favourable outcomes = 2 (5 and 6)
Total number of outcomes = 6
P(number > 4) = 2/6 = 1/3

Laws of probability

Probability theory is governed by some essential rules that help us calculate probabilities in different situations.

Rule 1: Sum up all probabilities

The sum of the probabilities of all possible outcomes in a sample space is 1.

P(outcome 1) + P(outcome 2) + ... + P(outcome n) = 1

For a six-sided dice, the rule is verified as follows:

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1

Law 2: Probability of an event not happening

The probability of an event not occurring is obtained by subtracting 1 from the probability of that event occurring.

P(not A) = 1 - P(A)

Example: Probability of not getting 6 on the dice:

P(6) = 1/6
P(not 6) = 1 - P(6) = 5/6

Rule 3: The sum rule

For any two events A and B, the probability of either A or B occurring is:

P(A or B) = P(A) + P(B) – P(A and B)

This rule ensures that each probability is calculated only once, even if the events overlap.

Rule 4: The multiplication rule

For independent events A and B, the probability that both A and B occur is:

P(A and B) = P(A) * P(B)

Conditional probability

Conditional probability is the probability of an event occurring, provided another event has already occurred. It is represented by P(A|B), which is read as "the probability of A given B."

The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

Conditional probability is mostly used in scenarios where events are dependent on each other.

Example: Conditional probability

Suppose you draw a card from a standard deck of 52 cards. You are interested in finding the probability that the card is a king, given that the card drawn is a spade.

Total cards: 52

Spades: 13 (because each suit has 13 cards)

King of Spades: There is only one King in Spades.

P(King | Spades) = P(King and Spades) / P(Spades)
P(king and spades) = 1/52
P(spade) = 13/52
P(King | Spades) = (1/52) / (13/52) = 1/13

Bayes' theorem

Bayes' theorem is a powerful result in probability that connects the conditional probability of two events. It tells us how to update the probabilities of hypotheses given evidence.

Bayes theorem formula:

P(A|B) = [P(B|A) * P(A)] / P(B)

This theorem is fundamental in a variety of fields, including statistics, finance, and machine learning.

Example: Using Bayes' theorem

Consider two boxes. Box 1 contains 3 red balls and 1 green ball, and box 2 contains 1 red ball and 2 green balls. A box is chosen at random, and the ball chosen at random is green. What is the probability that the box chosen was box 1?

Step 1: Define the events:

• A: The selected box is Box 1.
• B: The ball drawn is green.

Step 2: Find each possibility:

• P(A) = 1/2 (since the box can be either box 1 or box 2, with equal probability).
• P(B|A): If box 1 is chosen then probability of drawing a green ball = 1/4.
• P(B|A'): If box 2 is chosen then probability of drawing a green ball = 2/3.
• P(A'): 1 - P(A) = 1/2.

The probability P(B) can be calculated as:

P(B) = P(B|A) * P(A) + P(B|A') * P(A')
     = (1/4) * (1/2) + (2/3) * (1/2)
     = 1/8 + 1/3
     = 11/24

Use of Bayes' theorem:

P(A|B) = [P(B|A) * P(A)] / P(B)
       = [(1/4) * (1/2)] / (11/24)
       = 3/11

Conclusion

Probability theory provides us with a framework to analyze, predict, and extract meaning from random events by assigning numerical values to uncertain outcomes. We explored various fundamental concepts and rules in probability theory, including the basic definition of probability, types of events, laws of probability, conditional probability, and Bayes' theorem, along with illustrative examples to reinforce these concepts.

Through practice and application, probability can become a powerful tool in decision-making and prediction in many fields. Its applications are wide-ranging and help us deal with uncertainty in everyday life and in complex scientific fields.

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