Probability – - Addition Rule Of Probability Using Axiomatic Definition

Probability – Addition rule of Probability using Axiomatic definition

Probability is a branch of mathematics that deals with the likelihood of events occurring.
This topic focuses on the addition rule of probability using the axiomatic definition.
The axiomatic definition states that the probability of the union of two or more mutually exclusive events is the sum of their individual probabilities.
Let’s understand this concept with an example.

Example 1:

Consider a deck of playing cards.
What is the probability of drawing a red card or a face card? Solution:
We have two mutually exclusive events: drawing a red card and drawing a face card.
The probability of drawing a red card is 26/52 or 1/2.
The probability of drawing a face card is 12/52 or 3/13.
By applying the addition rule of probability, the probability of drawing a red card or a face card is (1/2) + (3/13) = 11/26.
The addition rule of probability can be extended to more than two events as well.
Let’s understand this concept further with another example. Example 2:
Consider rolling a fair six-sided die.
What is the probability of getting an odd number or a number greater than 4? Solution:
We have two mutually exclusive events: getting an odd number and getting a number greater than 4.
The probability of getting an odd number is 3/6 or 1/2.
The probability of getting a number greater than 4 is 2/6 or 1/3.
By applying the addition rule of probability, the probability of getting an odd number or a number greater than 4 is (1/2) + (1/3) = 5/6.
In some cases, the events may not be mutually exclusive.
Let’s explore the addition rule of probability for non-mutually exclusive events. Example 3:
Consider drawing a card from a deck.
What is the probability of drawing a heart or a face card? Solution:
The events “drawing a heart” and “drawing a face card” are not mutually exclusive, as the card can be both a heart and a face card.
We need to consider the overlap between these events.
The probability of drawing a heart is 13/52 or 1/4.
The probability of drawing a face card is 12/52 or 3/13.
To find the probability of drawing a heart or a face card, we subtract the probability of drawing a heart and a face card (which is also double-counted) from the sum of individual probabilities.
Therefore, the probability of drawing a heart or a face card is (1/4) + (3/13) - (1/26) = 15/26.
The addition rule of probability using the axiomatic definition is a fundamental concept in probability theory.
It helps us calculate the probability of events occurring together or separately.
It is applicable to mutually exclusive events as well as non-mutually exclusive events.
Understanding this concept is crucial for solving various probability problems in mathematics and real-life scenarios.
Let’s practice more examples to strengthen our understanding.

Probability – Addition rule of Probability using Axiomatic definition

Mutually Exclusive Events:

Event A and Event B are said to be mutually exclusive if they cannot both occur at the same time.
In other words, if Event A occurs, then Event B cannot occur, and vice versa.
For mutually exclusive events, the addition rule of probability is straightforward.
The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
Mathematically, P(A or B) = P(A) + P(B).

Example:

Consider rolling a fair six-sided die.
What is the probability of getting a 2 or a 4? Solution:
The events “getting a 2” and “getting a 4” are mutually exclusive.
The probability of getting a 2 is 1/6.
The probability of getting a 4 is also 1/6.
By applying the addition rule of probability, the probability of getting a 2 or a 4 is 1/6 + 1/6 = 1/3.

Non-Mutually Exclusive Events:

Event A and Event B are said to be non-mutually exclusive if they can both occur at the same time.
In such cases, we need to consider the overlap between the events.
The addition rule of probability for non-mutually exclusive events is as follows:
- P(A or B) = P(A) + P(B) - P(A and B).

Example:

Consider drawing a playing card from a standard deck.
What is the probability of drawing a heart or a king? Solution:
The events “drawing a heart” and “drawing a king” are not mutually exclusive.
The probability of drawing a heart is 1/4.
The probability of drawing a king is 4/52.
To find the probability of drawing a heart or a king, we subtract the probability of drawing a king of hearts (which is double-counted) from the sum of individual probabilities.
Therefore, the probability of drawing a heart or a king is (1/4) + (4/52) - (1/52) = 15/52.

General Addition Rule:

The addition rule of probability can be extended to more than two events.
For any two events A and B, the probability of their union is given by:
- P(A or B) = P(A) + P(B) - P(A and B).
Similarly, the probability of the union of three events A, B, and C can be calculated using:
- P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C).

Example:

Consider a bag containing red, blue, and green balls.
What is the probability of drawing a red or a blue ball or a green ball? Solution:
Suppose the probability of drawing a red ball is 3/10.
The probability of drawing a blue ball is 2/10.
The probability of drawing a green ball is 5/10.
The probability of drawing a red and blue ball is 1/10.
The probability of drawing a red and green ball is 2/10.
The probability of drawing a blue and green ball is 3/10.
By applying the general addition rule of probability, the probability is (3/10) + (2/10) + (5/10) - (1/10) - (2/10) - (3/10) = 14/10.

Complementary Events:

Complementary events are mutually exclusive events whose probabilities sum up to 1.
Mathematically, if A is an event, then the probability of A plus the probability of not A is 1.
P(A) + P(not A) = 1.
The probability of not A is denoted as P(A’).

Example:

Consider tossing a fair coin.
What is the probability of getting heads or tails? Solution:
Let’s say the probability of getting heads is P(H) = 1/2.
The probability of getting tails is P(T) = P(H’) = 1/2.
By applying the addition rule of probability, the probability of getting heads or tails is 1/2 + 1/2 = 1.

Addition Rule for Mutually Exclusive Events:

Mutually exclusive events have no common outcomes.
If A and B are mutually exclusive, then P(A and B) = 0.
In this case, the addition rule of probability simplifies to:
- P(A or B) = P(A) + P(B).

Summary:

The addition rule of probability using the axiomatic definition states that the probability of the union of two or more mutually exclusive events is the sum of their individual probabilities.
For non-mutually exclusive events, we need to subtract the probability of their overlap to avoid double counting.
The general addition rule of probability can be used for more than two events.
Complementary events have probabilities that sum up to 1.
Understanding the addition rule of probability is essential for solving probability problems and analyzing real-world scenarios.

Conditional Probability:

Conditional probability is the probability of an event occurring given that another event has already occurred.
It is denoted as P(A | B), read as “the probability of A given B.”
Mathematically, P(A | B) = P(A and B) / P(B).
This formula allows us to calculate the probability of A given that B has already occurred.

Example:

Consider drawing two cards from a deck without replacement.
What is the probability of drawing a red card on the second draw, given that the first card drawn was red? Solution:
Let’s assume that the first card drawn was red.
The probability of drawing a red card on the second draw is the probability of drawing a red card and a red card on both draws divided by the probability of the first card being red.
The probability of drawing a red card and a red card is (26/52) * (25/51).
The probability of the first card being red is 26/52.
By applying the formula, we can find the conditional probability: P(red on 2nd draw | red on 1st draw) = ((26/52) * (25/51)) / (26/52) = 25/51.

Independence of Events:

Two events A and B are said to be independent if the probability of one event occurring does not affect the probability of the other event occurring.
Mathematically, if A and B are independent events, then P(A | B) = P(A) and P(B | A) = P(B).
This means that the occurrence of one event does not provide any information about the occurrence of the other event.

Example:

Consider rolling two fair six-sided dice.
What is the probability of getting a 4 on the first die and a 3 on the second die? Solution:
The probability of getting a 4 on the first die is 1/6.
The probability of getting a 3 on the second die is also 1/6.
Since the rolls of the dice are independent, the probability of both events happening is: P(4 on 1st die and 3 on 2nd die) = (1/6) * (1/6) = 1/36.

Multiplication Rule for Independent Events:

The multiplication rule for independent events states that the probability of two independent events occurring together is the product of their individual probabilities.
Mathematically, if A and B are independent events, then P(A and B) = P(A) * P(B).
This rule can be extended to more than two independent events as well.

Example:

Consider flipping a fair coin twice.
What is the probability of getting heads on both flips? Solution:
The probability of getting heads on the first flip is 1/2.
The probability of getting heads on the second flip is also 1/2.
By applying the multiplication rule for independent events, the probability of getting heads on both flips is: P(heads on 1st flip and heads on 2nd flip) = (1/2) * (1/2) = 1/4.

Total Probability Theorem:

The total probability theorem is used to find the probability of an event A given that another event B has occurred.
It states that the probability of event A is the sum of the probabilities of A occurring in each possible way, considering all mutually exclusive and exhaustive events.

Example:

Consider a bag containing 3 red balls and 5 blue balls.
A ball is drawn at random.
It is known that the ball is either red or blue.
What is the probability of the ball being red? Solution:
Let A be the event of drawing a red ball and B be the event of drawing either a red or blue ball.
The probability of drawing a red ball given that the ball is either red or blue can be calculated using the total probability theorem.
P(red) = P(red and B1) + P(red and B2)
Here, B1 represents drawing a red ball, and B2 represents drawing a blue ball.
P(red and B1) = (3/8)
P(red and B2) = 0
Therefore, P(red) = (3/8) + 0 = 3/8.

Bayes’ Theorem:

Bayes’ theorem is used to calculate the probability of an event A given that another event B has occurred.
It is expressed as:
- P(A | B) = (P(B | A) * P(A)) / P(B)
- P(A) and P(B) are the probabilities of A and B occurring independently.
- P(B | A) is the probability of B occurring given A has occurred.
- P(A | B) is the probability of A occurring given B has occurred.

Example:

Consider a factory that produces light bulbs.
The factory produces two types of light bulbs: Type A and Type B.
It is known that 10% of all produced bulbs are defective.
It is also known that 8% of Type A bulbs are defective, while 12% of Type B bulbs are defective.
What is the probability that a randomly selected defective bulb is of Type A? Solution:
Let A be the event of selecting a Type A bulb, and B be the event of selecting a defective bulb.
P(A) = 0.5 (assuming an equal number of Type A and Type B bulbs are produced)
P(B) = 0.1 (given that 10% of all bulbs are defective)
P(B | A) = 0.08 (given that 8% of Type A bulbs are defective)
By applying Bayes’ theorem, we can calculate P(A | B):
- P(A | B) = (P(B | A) * P(A)) / P(B)
- P(A | B) = (0.08 * 0.5) / 0.1
- P(A | B) = 0.04 / 0.1
- P(A | B) = 0.4