Axioms of Probability - Probability and Statistics | Engineering Mathematics | GATE CSE

Summary

This session explores the fundamentals of probability, discussing the probability of events being greater than zero but less than one, the certainty of sample spaces at 100%, the addition rule for non-mutually exclusive events, and the multiplication rule for dependent events. It also covers conditional probability, changing the sample space based on conditions, and the concept of independent events where probabilities multiply directly.

Takeaways

• 😀 The probability of an event E in a sample space S ranges from greater than or equal to zero to less than or equal to one, and in percentage terms, it's between 0% to 100%.
• 📚 The probability of the entire sample space is always one or 100%, as it represents all possible outcomes.
• 🔍 For any two events A and B, the probability of their intersection can be calculated using the formula P(A ∩ B) = P(A) + P(B) - P(A ∩ B).
• 🔄 If events A and B are mutually exclusive, their intersection probability P(A ∩ B) becomes zero, and the union probability is simply the sum of their individual probabilities P(A ∪ B) = P(A) + P(B).
• 🤔 The multiplication rule states that the probability of the intersection of events A and B is P(A ∩ B) = P(A) * P(B | A) or P(A) * P(B given A), provided P(A) ≠ 0 and P(B) ≠ 0.
• 🎲 Conditional probability changes the sample space, focusing on the outcomes where a certain condition has already occurred, such as the probability of rolling a 3 given that the roll is an odd number.
• 🚫 If events A and B are independent, the probability of their intersection is simply the product of their individual probabilities, P(A ∩ B) = P(A) * P(B).
• 🔗 The concept of conditional probability is crucial in understanding how the probability of one event changes given the occurrence of another event.
• 📉 The sample space can be reduced in conditional probability scenarios, as seen in the example where the sample space is reduced to odd numbers when calculating the probability of rolling a 3 given that the roll is odd.
• 🔄 For multiple independent events, the probability of their intersection is the product of their individual probabilities, extending the rule for two events to any number of events.

Q & A

What is the range of probability values for an event E in the sample space S?

The probability of an event E is greater than or equal to 0 and less than or equal to 1, which can also be expressed as between 0% to 100%.

Why is the probability of the sample space always 100%?

The sample space represents all possible outcomes of an experiment, ensuring at least one outcome will definitely occur, thus covering the entire space and making its probability 100%.

How is the probability of the intersection of two events A and B calculated?

The probability of the intersection of A and B is given by the sum of the probabilities of A and B minus the probability of their intersection, i.e., P(A ∩ B) = P(A) + P(B) - P(A ∩ B).

What happens to the probability of the intersection of two mutually exclusive events A and B?

If events A and B are mutually exclusive, there is nothing in common between them, making the probability of their intersection zero, P(A ∩ B) = 0.

How does the probability of the union of mutually exclusive events differ from non-mutually exclusive events?

For mutually exclusive events, the probability of their union is the sum of their individual probabilities, P(A ∪ B) = P(A) + P(B), as there is no overlap.

What is the multiplication rule for the probability of two events A and B?

The multiplication rule states that the probability of the intersection of A and B is the product of the probability of A and the probability of B given that A has occurred, or vice versa, P(A ∩ B) = P(A) * P(B|A) or P(A ∩ B) = P(B) * P(A|B), provided P(A) ≠ 0 and P(B) ≠ 0.

What is the difference between joint probability and marginal probability?

Joint probability refers to the probability of two events occurring together, like P(A ∩ B). Marginal probability refers to the probability of a single event occurring, such as P(A) or P(B), without considering the other event.

How does conditional probability change the sample space?

Conditional probability, such as P(A|B), changes the sample space by considering only those outcomes where the condition (B has occurred) is met, effectively reducing the sample space to only those outcomes that satisfy the condition.

Can you provide an example of conditional probability from the script?

An example given in the script is the probability of rolling a 3 given that the roll is an odd number. The new sample space is reduced to the odd numbers {1, 3, 5}, making the probability of rolling a 3 1/3.

What is the relationship between the probability of two independent events and their intersection?

For independent events A and B, the probability of their intersection is the product of their individual probabilities, P(A ∩ B) = P(A) * P(B), as the occurrence of one event does not affect the other.

How does the probability of the intersection of multiple independent events extend the rule for two events?

For n independent events, the probability of their intersection is the product of their individual probabilities, P(E1 ∩ E2 ∩ ... ∩ En) = P(E1) * P(E2) * ... * P(En), without any additional requirements.

Outlines

00:00

📊 Fundamentals of Probability

This paragraph introduces the basic concepts of probability theory. It starts by defining the sample space (S) and an event (E) within it, stating that the probability of an event is always between 0 and 1, inclusive. It also highlights that the probability of the entire sample space is 100%, as it encompasses all possible outcomes. The paragraph further explains the addition rule for the probability of the union of two events, E and B, and how it simplifies when E and B are mutually exclusive. It concludes with the multiplication rule for the probability of the intersection of E and B, given that neither event has a zero probability, introducing the concept of conditional probability.

05:04

🎲 Conditional Probability and Independence

The second paragraph delves into conditional probability, explaining how it changes the sample space based on given conditions. It uses the example of rolling a die to illustrate how the probability of rolling a 3 changes when the condition is that the number rolled is odd. The new sample space is reduced to odd numbers, and the probability of rolling a 3 under this condition is calculated. The paragraph also discusses the concept of independent events, where the occurrence of one event does not affect the probability of another. It explains that for independent events, the probability of their intersection is the product of their individual probabilities. The explanation extends to multiple independent events, showing that the probability of their intersection is the product of their individual probabilities without any additional requirements.

Keywords

💡Sample Space (S)

The sample space refers to the set of all possible outcomes of an experiment. It is foundational in understanding probability as it encompasses every possible result that could occur. In the video, the sample space is mentioned as having a probability of one or 100%, indicating that at least one outcome from this set will definitely occur, covering the entire scope of possible events.

💡Event (E)

An event is a specific outcome or a combination of outcomes from the sample space that we are interested in for a particular probability calculation. The video script discusses how the probability of an event is greater than or equal to zero and less than or equal to one, highlighting the fundamental range within which all event probabilities must fall.

💡Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. The script explains that probability quantifies the chance of an event happening, with 0 indicating impossibility and 1 indicating certainty, and it is central to the theme of the video.

💡Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time; they have no outcomes in common. The script uses this concept to explain that when events A and B are mutually exclusive, the probability of their intersection is zero, simplifying the calculation of their union to the sum of their individual probabilities.

💡Intersection

The intersection of two events A and B refers to the outcomes that are common to both events. The video script explains the formula for the probability of the intersection, which is a fundamental concept in probability theory, especially when discussing whether events are mutually exclusive or not.

💡Union

The union of events A and B includes all outcomes that are part of either event A, event B, or both. The script clarifies that for mutually exclusive events, the probability of the union is simply the sum of the probabilities of the individual events, which is a key point in understanding combined probabilities.

💡Conditional Probability

Conditional probability is the probability of an event given that another event has already occurred. The video provides the formula for conditional probability and explains how it changes the sample space, using the example of the probability of rolling a 3 given that the roll is an odd number, which alters the sample space to only include odd numbers.

💡Marginal Probability

Marginal probability refers to the probability of a single event occurring by itself, without any conditions. The script contrasts this with joint and conditional probabilities, emphasizing that marginal probabilities do not take into account the occurrence of other events.

💡Multiplication Rule

The multiplication rule is a fundamental principle in probability that allows for the calculation of the probability of two events both occurring. The video script describes this rule, stating that the probability of the intersection of events A and B is the product of the probability of A and the probability of B given A has occurred, provided neither probability is zero.

💡Independence

Independence in the context of events means that the occurrence of one event does not affect the probability of the occurrence of another. The script explains that for independent events, the probability of the intersection is the product of their individual probabilities, simplifying the calculation and highlighting a key concept in probability theory.

💡Joint Probability

Joint probability is the probability that two or more events will occur together. The video script discusses how to calculate joint probability using the multiplication rule, emphasizing its importance in understanding the relationship between events.

Highlights

Probability of an event E is greater than or equal to 0 and less than or equal to 1.

Probability of the sample space S is always 100% as it includes all possible outcomes.

For events A and B, the probability of their intersection can be calculated using the formula P(A ∩ B) = P(A) + P(B) - P(A ∩ B).

If A and B are mutually exclusive, P(A ∩ B) becomes 0, simplifying the union probability to P(A ∪ B) = P(A) + P(B).

The sum of probabilities for any number of mutually exclusive events is a straightforward addition of their individual probabilities.

The multiplication rule for events A and B is given by P(A ∩ B) = P(A) * P(B|A) or P(B) * P(A|B), assuming P(A) and P(B) are not zero.

Conditional probability, P(B|A), changes the sample space based on the condition that event A has already occurred.

An example of conditional probability is the probability of rolling a 3 given the roll is an odd number, which reduces the sample space to odd numbers only.

The new sample space for the conditional probability of getting a 3 given an odd roll is {1, 3, 5}, changing the probability calculation.

For independent events, the occurrence of one does not affect the probability of the other, simplifying the intersection probability to P(A ∩ B) = P(A) * P(B).

The concept of extending the multiplication rule to n independent events, where the probability is the product of their individual probabilities.

Marginal probability refers to the probability of a single event occurring, as opposed to joint or conditional probabilities.

The importance of understanding the difference between joint, marginal, and conditional probabilities in probability theory.

The impact of conditional probability on the sample space, exemplified by the change in sample space when considering the probability of getting a 3 given an odd roll.

The practical application of probability rules in determining the likelihood of events in various scenarios, such as dice rolls.

The session provides a comprehensive overview of fundamental probability concepts and their applications.

The session emphasizes the importance of correctly identifying and calculating probabilities in different types of events.

Transcripts

hello everyone in this session we'll

discuss exams of probability in this

session we'll see exams of probability

so let's start with the let's say that s

is the sample space and E is the event

then probability of event is greater

than zero equal to and sorry less and

greater than equal to zero and less than

equal to one if we are talking in

percentage it is going to be between

zero to hundred percentage zero to

hundred percent secondly the probability

of sample spaces one or one hundred

percent because the sample space is a

set of all the possible outcomes and

probability of sample space that means

at least one of the outcome will

definitely occur and then it will cover

the entire space so it is 100 percent

third is let's say we have events E and

B for any foreign experiment let's say

we have events a and B then probability

of a intersection B can be given as

probability of a plus probability of B

minus probability of a intersection B

right will add the probability of a B

and then subtract the a intersection B

and what if a and B are mutually

exclusive if they are mutually exclusive

this a into

Section B will become zero disjoint a

and B are disjoint there is nothing in

common between a and B so in that case

in that case probability of a union B

directly becomes probability of a plus

probability of B for any number of

mutually exclusive events P of let's say

even in the section e to intersection up

to e n can be directly written as

probability of even plus probability of

e 2 plus probability of e n sum of all

sum of all the probabilities right

fourth one is let's say for a and B are

events of random experiment then

probability of a intersection B is given

as probability of a into probability of

B given a has occurred or probability of

B into probability of a given B and this

is known as multiplication rule

condition P of a is not equal to zero

and P of B is not equal to zero

so what does this be given a this is a

conditional probability in short if we

see this a intersection B is a joint

probability probability of a or

probability B alone is marginal

probability because it is about one

single event and this is conditional

probability probability of B conditioned

a has already occurred probability of a

condition from B has already occurred

right so a little detail on this

conditional probability will help you

this probability of a given B actually

changes the sample space it reduces the

sample space for example let's say

probability I am saying I'm giving an

example probability of getting 3 given

the number is odd I'm talking about a

dice so probability of getting a 3 given

the number is odd now in general the

entire sub spare entire events entire

sample space is considered so the number

of events in the sample space is 6 but

here the condition is the given the

number is odd so the sample space

reduces to only odd numbers so we have

odd numbers as 1 3 & 5 right 1 3 & 5 so

this becomes the new sample space let's

say s - and getting a three is your

event right given it is odd so we don't

have the entire sample space as one two

three four five six

we have only odd numbers 1 3 5 so the

number of event is 3 sample space is 3

and number of event is 1 so the

probability becomes 1 by 3 it is not 1

by 6 anymore it is a conditional

probability so it will be 1 by 3 now

right

and also if if a and B are independent

we have read in types of events that for

an event being independent it is

directly equal to the probability of

that event that means this P probability

of B given a will become probability of

B and probability of a given B will

directly become probability of a because

it is independent of the occurrence of a

or it is independent of the occurrence

of B so in that case if these are

independent probability of a

intersection B is directly given as

probability of a into probability of B

we can add or we can write this as since

probability of a will become will be

equal to probability of a given B and

probability of B will be equal to

probability of B given a for independent

events

right and of course like the previous

one we're gonna extend this so for

number let's say n numbers of

independent event probability of even

intersection a 2 intersection III

intersection E n will directly become

probability of even into probability of

e 2 into da da dot up to probability of

e em you can directly multiply it

without any other requirement

you