Axioms of Probability - Probability and Statistics | Engineering Mathematics | GATE CSE
Summary
TLDRThis 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
📊 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.
🎲 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.
Mindmap
Keywords
💡Sample Space (S)
💡Event (E)
💡Probability
💡Mutually Exclusive Events
💡Intersection
💡Union
💡Conditional Probability
💡Marginal Probability
💡Multiplication Rule
💡Independence
💡Joint Probability
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
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