Probability
Probability is the branch of mathematics that is concerned with the chances of occurrence of events and possibilities. Further, it gives a method of measuring the probability of uncertainty and predicting events in the future by using the available information.
Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event). The probability of an event
A is often written as P(A).
Basic Concepts
- Experiment: An action or process that leads to one or more outcomes. For example, tossing a coin.
- Sample Space (S): The set of all possible outcomes of an experiment. For a coin toss,
S={Heads, Tails}.
- Event: A subset of the sample space. For instance, getting a head when tossing a coin.
Formula for Probability
The probability of an event A is given by:
P(A)= Number of favorable outcomes / Total number of possible outcomes
Axioms of Probability
There are three axioms that are the basis of probability and are as follows:
- Non-Negativity: For any event A, the probability of A is always non-negative:
P(A)\geq0 - Normalization: The total chance of the whole possible outcomes of the sample space S:
P(S)=1 - Additivity: For any two mutually exclusive events we have A and B (i.e., events that cannot occur simultaneously), the probability of their union is the sum of their individual probabilities:
P(A∪B)=P(A)+P(B)if A∩B=0
Properties of Probability
Properties of Probability: Probability is a branch of mathematics that specifies how likely an event can occur. The value of probability is between 0 and 1. Zero(0) indicates an impossible event and One(1) indicates certainly (surely) that will happen. There are a few properties of probability that are mentioned below:
Key Properties of Probability:
- Non-Negativity: The probability of any event is always non-negative. For any event A,
P(A) ≥ 0.
- Normalization: The probability of the sure event (sample space) is 1. If S is the sample space, then
P(S) = 1.
- Additivity (Sum Rule): For any two mutually exclusive (disjoint) events A and B, the probability of their union is the sum of their individual probabilities:
P(A∪B) = P(A) + P(B)
- Complementary Rule: The probability of the complement of an event A (i.e., the event not occurring) is
P(A∣B)= P(A∩B) / P(B)
provided P(B)>0.
- Multiplication Rule: For any two events A and B, the probability of both occurring (i.e., the intersection) is:
P(A∩B) = P(A∣B) ⋅ P(B)
1. The probability of an event can be defined as the number of favorable outcomes of an event divided by the total number of possible outcomes of an event.
Probability(Event) = (Number of favorable outcomes of an event) / (Total Number of possible outcomes).
Example: What is the probability of getting a Tail when a coin is tossed?
Solution:
Number of Favorable Outcomes- {Tail} = 1
Total Number of possible outcomes- {Head, Tail} - 2
Probability of getting Tail= 1/2 = 0.5
2. Probability of a sure/certain event is 1.
Example: What is the probability of getting a number between 1 and 6 when a dice is rolled?
Solution:
Number of favorable outcomes- {1,2,3,4,5,6} = 6
Total Possible outcomes- {1,2,3,4,5,6} = 6
Probability of getting a number between 1 to 6= 6/6 = 1
Probability is 1 indicates it is a certain event.
3. The probability of an impossible event is zero (0).
Example: What is the probability of getting a number greater than 6 when a dice is rolled?
Solution:
Number of favorable outcomes - {} = 0
Total possible outcomes - {1,2,3,4,5,6} = 6
Probability(Number>6) = 0/6 = 0
Probability Zero indicates impossible event.
4. Probability of an event always lies between 0 and 1. It is always a positive.
0 <= Probability(Event) <= 1
Example: We can notice that in all the above examples probability is always between 0 & 1.
5. If A and B are 2 events that are said to be mutually exclusive events then P(AUB) = P(A) + P(B).
Note: Two events are mutually exclusive when if 2 events cannot occur simultaneously.
Example: Probability of getting head and tail when a coin is tossed comes under mutual exclusive events.
Solution:
To solve this we need to find probability separately for each possibility.
i.e, Probability of getting head and Probability of getting tail and sum of those to get P(Head U Tail).
P(Head U Tail)= P(Head) + P(Tail) = (1/2)+(1/2) = 1
6. Elementary event is an event that has only one outcome. These events are also called atomic events or sample points. The Sum of probabilities of such elementary events of an experiment is always 1.
Example: When we are tossing a coin the possible outcome is head or tail. These individual events i.e. only head or only tail of a sample space are called elementary events.
Solution:
Probability of getting only head=1/2
Probability of getting only tail=1/2
So, sum=1.
7. Sum of probabilities of complementary events is 1.
P(A)+P(A')=1
Example: When a coin is tossed, the probability of getting ahead is 1/2, and the complementary event for getting ahead is getting a tail so the Probability of getting a tail is 1/2.
Solution:
If we sum those two then,
P(Head)+P(Head')=(1/2)+(1/2)=1
Head'= Getting Tail
8. If A and B are 2 events that are not mutually exclusive events then
- P(AUB)=P(A)+P(B)-P(A∩B)
- P(A∩B)=P(A)+P(B)-P(AUB)
Note: 2 events are said to be mutually not exclusive when they have at least one common outcome.
Example: What is the probability of getting an even number or less than 4 when a die is rolled?
Solution:
Favorable outcomes of getting even number ={2,4,6}
Favorable outcomes of getting number<4 ={1,2,3}
So, there is only 1 common outcome between two events so these two events are not mutually exclusive.
So, we can find P(Even U Number<4)= P(Even) + P(Number<4) - P(Even ∩ Number<4)
P(Even)=3/6=1/2
P(Number<4)=3/6=1/2
P(Even ∩ Number<4)=1/6 (Common element)
P(Even U Number<4)=(1/2) +(1/2)-(1/6)=1-(1/6)=0.83
9. If E1,E2,E3,E4,E5,.........EN are mutually exclusive events then Probability(E1UE2UE3UE4UE5U......UEN)=P(E1)+P(E2)+P(E3)+P(E4)+P(E5)+.......+P(EN).
Example: What is the probability of getting 1 or 2 or 3 numbers when a die is rolled.
Solution:
Let A be the event of getting 1 when a die is rolled.
Favorable outcome- {1}
Let B be the event of getting 2 when a die is rolled.
Favorable outcome- {2}
Let C be the event of getting 3 when a die is rolled.
Favorable outcome- {3}
No common favorable outcomes.
So, A, B, C are mutually exclusive events.
According to above probability rule- P(A U B U C)= P(A) + P(B) + P(C)
P(A)=1/6
P(B)=1/6
P(C)=1/6
P(A U B U C)=(1/6)+(1/6)+(1/6)=3/6=1/2
These are the top basic properties of probability.
Conditional Probability
Conditional probability quantifies the probability of an event A given that another event B has occurred. It is defined as:
P(A∣B)= P(A∩B)/P(B)
,provided P(B)>0
Law of Total Probability
The law of total probability enables us the formulation in terms of probabilities of an event A as a weighted sum of conditional probabilities:
P(A)= \Sigma_{i=0}^{n}
P(A|B_i)P(B_i)
where {B_1,B_2,...,B_n} is a partition of sample space S.
Bayes' Theorem
Bayes' Theorem provides a way to update the probability of a hypothesis H based on new evidence E:
P(H∣E)= P(E∣H)P(H)/P(E)
where P(H) is the prior probability of the hypothesis, P(E∣H) is the likelihood of the evidence given the hypothesis, and P(E) is the marginal likelihood of the evidence.
Independence of Events
Two events A and B are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other:
P(A∩B)=P(A)⋅P(B)
If P(A\cap B) \neq P(A)P(B) the events are dependent.
Random Variables and Expectation
A random variable is a variable that takes values that are at the same time both random and mathematical in a given sample space. There are two main types of random variables:There are two main types of random variables:
- Discrete Random Variables: Be limited or countable in the sense that they take on one of a finite or countably infinite number of values.
- Continuous Random Variables: Thus theses converge on beavering on an uncountably infinite set of values.
Expectation of a Random Variable
The expectation (or expected value) of a random variable X, denoted by E(X), is a measure of the central tendency of its distribution.
- For a discrete random variable X with possible values x_1,x_2,... and corresponding probabilities p_1,p_2,...
E(X)= ∑_i x_i⋅P(X=x_i) - For a continuous random variable X with probability density function f(x):
E(X)=\int_{-\infty }^{\infty }x⋅f(x)dx
Variance and Standard Deviation
Variance measures the spread of a random variable around its expectation and is denoted by Var(X). The standard deviation is the square root of the variance, denoted by \sigma_X.
- Variance of a random variable X:
Var(X)=E[(X−E(X))^2
]=E(X^2
)−[E(X)]^2 - Standard Deviation of X:
σ_X = Var(X)
Relationship with Expectation
The variance can also be understood as the mathematical expectation of the value of squares of the deviations from the mean, which provides an understanding of the dispersion of the values of the random variable.
Probability Distributions
This basically gives the manner in which probabilities are spread over the values of the random variable involved. Some common distributions include:
- Discrete Distributions:
- Binomial Distribution
- Poisson Distribution
- Continuous Distributions:
- Normal Distribution
- Exponential Distribution
Properties of Probability - Sample Problems
Example 1: A fair die is rolled. What is the probability of rolling a number greater than 6?
Solution: Since there are no numbers greater than 6 on a standard die, P(rolling > 6) = 0
Example 2: A coin is tossed. Verify that the sum of probabilities of all outcomes is 1.
Solution: P(Heads) = 1/2, P(Tails) = 1/2
1/2 + 1/2 = 1, so the property holds.
Example 3 : In a deck of 52 cards, what is the probability of drawing either a king or a queen?
Solution: P(King) = 4/52 = 1/13, P(Queen) = 4/52 = 1/13
P(King or Queen) = 1/13 + 1/13 = 2/13
Example 4: The probability of rain tomorrow is 0.3. What is the probability it won't rain?
Solution: P(no rain) = 1 - P(rain) = 1 - 0.3 = 0.7
Example 5: In a standard deck, compare P(drawing a king) and P(drawing a face card).
Solution: P(King) = 4/52 = 1/13
P(Face card) = 12/52 = 3/13
Since all kings are face cards, P(King) ≤ P(Face card)
Example 6: In a class, 60% of students play soccer, 30% play basketball, and 20% play both. What percentage plays either soccer or basketball?
Solution: P(Soccer or Basketball) = P(Soccer) + P(Basketball) - P(Soccer and Basketball)
= 0.60 + 0.30 - 0.20 = 0.70 or 70%
Example 7: A fair coin is tossed twice. What's the probability of getting heads both times?
Solution: P(H on first toss) = 1/2, P(H on second toss) = 1/2
P(H and H) = 1/2 × 1/2 = 1/4
Example 8: In a deck of 52 cards, what's the probability of drawing a king, given that it's a face card?
Solution: P(King | Face card) = P(King and Face card) / P(Face card)
= (4/52) / (12/52) = 1/3
Example 9: 30% of students are in Science. 80% of Science students and 60% of non-Science students wear glasses. What percentage of all students wear glasses?
Solution: P(Glasses) = P(Glasses|Science) × P(Science) + P(Glasses|not Science) × P(not Science)
= 0.80 × 0.30 + 0.60 × 0.70 = 0.24 + 0.42 = 0.66 or 66%
Example 10: 1% of people have a certain disease. The test for this disease is 95% accurate (both for positive and negative results). If a person tests positive, what's the probability they have the disease?
Solution: Let D = disease, T = positive test
P(D|T) = [P(T|D) × P(D)] / [P(T|D) × P(D) + P(T|not D) × P(not D)]
= (0.95 × 0.01) / (0.95 × 0.01 + 0.05 × 0.99)
≈ 0.1611 or about 16.11%
Practice Problems on Properties of Probability
1).A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn at random, what is the probability of drawing either a red or a green marble?
2).In a class of 30 students, 18 play football, 15 play basketball, and 10 play both. What is the probability that a randomly selected student plays neither sport?
3).The probability of a student passing a math test is 0.7, and the probability of passing a science test is 0.8. If the events are independent, what is the probability of passing both tests?
4).A fair six-sided die is rolled. What is the probability of rolling an even number or a number greater than 4?
5).In a deck of 52 cards, what is the probability of drawing a heart, given that the card drawn is red?
6).The probability of it raining on any given day is 0.3. What is the probability that it will rain on at least one day during a 3-day period? (Assume independence)
7).60% of a company's employees are women. 30% of the employees work in IT. If 20% of the women work in IT, what percentage of IT employees are men?
8).A test for a certain disease is 95% accurate for positive results and 98% accurate for negative results. If 2% of the population has this disease, what is the probability that a person who tests positive actually has the disease?
9).Box A contains 3 red balls and 2 blue balls. Box B contains 2 red balls and 5 blue balls. If a box is chosen at random and then a ball is drawn from it, what is the probability of drawing a red ball?
10).The probability of a student being left-handed is 0.1. In a class of 20 students, what is the probability that exactly 2 students are left-handed?
Applications of Probability
1. Risk Management
Probability is used in risk management to assess the likelihood of adverse events and to develop strategies to mitigate their impact.
2. Quality Control
In manufacturing, probability helps in quality control processes to determine the likelihood of defects and to improve production methods.
3. Decision Making
Probability aids in decision making by providing a framework for evaluating different options based on their potential outcomes.
4. Game Theory
Probability is essential in game theory for analyzing strategies and predicting the behavior of competitors in various scenarios.
5. Engineering
In engineering, probability is used in reliability analysis to evaluate the performance and safety of systems and components.
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Solved Examples Properties of Probability
Question: What is the probability of drawing an Ace from a standard deck of 52 playing cards?
Solution:
Number of favorable outcomes (Aces) = 4
Total number of possible outcomes (cards) = 52
Using the probability formula:
P(Ace)=Number of favorable outcomes/Total number of possible outcomes = 4/52 = 1/13.
Therefore, the probability of drawing an Ace is 1/13.
Question: What is the probability of rolling a number between 1 and 6 on a fair six-sided die?
Solution:
Number of favorable outcomes = 6 (since all outcomes 1, 2, 3, 4, 5, 6 are between 1 and 6)
Total number of possible outcomes = 6
Using the probability formula:
P(Number between 1 and 6)=Number of favorable outcomes / Total number of possible outcomes = 6 / 6 = 1
Therefore, the probability of rolling a number between 1 and 6 is 1 (a certain event).
Summary - Properties of Probability
Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood of events occurring. It is based on several key properties and rules that govern how probabilities are calculated and interpreted. These include the basic property that probabilities range from 0 (impossible event) to 1 (certain event), and that the sum of probabilities for all possible outcomes in a sample space equals 1. Other important principles include the addition rule for mutually exclusive events, the multiplication rule for independent events, conditional probability, and Bayes' theorem. These properties allow for the analysis of complex scenarios involving multiple events, dependent outcomes, and updated probabilities based on new information. Understanding and applying these probability properties is crucial in various fields, including science, engineering, finance, and decision-making, as they provide a framework for quantifying uncertainty, assessing risks, and making predictions based on available data. Mastery of these concepts enables more accurate analysis of real-world situations and informed decision-making in the face of uncertainty.
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