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conditional probability and bayes theorem for gate
Conditional probability and Bayes' Theorem are fundamental concepts in probability theory, often tested in the GATE exam. They help us understand how the likelihood of an event changes when we have additional information. Mastering these concepts is crucial for solving complex probability and statistics problems.
Conditional probability is a concept in probability theory that allows us to assess the likelihood of one event occurring, given that another event has already taken place. It's a way to refine our probability estimates based on additional information.
Conditional probability plays a vital role in various fields, including statistics, finance, healthcare, and machine learning. It's essential because it enables us to make more accurate predictions and decisions by incorporating context or prior knowledge.
In conditional probability notation, P(A|B) represents the probability of event A happening given that event B is known to have occurred. This notation helps us explicitly define and calculate probabilities under specific conditions.
Understanding conditional probability serves as the foundation for exploring Bayes' Theorem. Bayes' Theorem extends the concept of conditional probability and is especially valuable in situations involving uncertainty and incomplete information.
1. Conditional Probability Formula:
The conditional probability formula is expressed as:
P(A|B) = P(A ∩ B) / P(B)
In this formula:
The formula is valuable in real-world scenarios where we want to refine our predictions based on new information. For instance, if we're diagnosing a medical condition (A) and we know the results of a specific test (B), the conditional probability formula helps us estimate the probability of the condition given the test results.
Suppose you are managing an e-commerce website, and you want to understand how the behavior of users changes depending on their device type. You collect data on two events:
You want to calculate the conditional probability of a user making a purchase (Event A) given that they visited the website from a mobile device (Event B).
To calculate P(A|B), we'll use the conditional probability formula:
P(A|B) = P(A ∩ B) / P(B)
1. Calculate P(A ∩ B): This is the probability that a user both makes a purchase and visits the website from a mobile device.
2. Calculate P(B): This is the probability that a user visits the website from a mobile device.
3. Now, plug these values into the formula:
P(A_∣_B) = P(A_∩_B)/P(B) = 0.2/0.5=0.4
Suppose you have data from a survey of customers at a coffee shop. You are interested in understanding the probability of a customer ordering an espresso (Event A) given that they also ordered a pastry (Event B). You have the following information:
Using conditional probability, calculate the probability that a customer who ordered a pastry also ordered an espresso (P(A|B)).
We want to find the probability that a customer ordered an espresso (Event A) given that they ordered a pastry (Event B), which is represented as P(A|B).
We are given the following information:
In probability theory, two events are considered independent if the occurrence of one event does not affect the occurrence or non-occurrence of the other. In other words, the probability of both events happening together is simply the product of their individual probabilities. Mathematically, two events A and B are independent if:
Here, P(A_∩_B) represents the probability of both events A and B occurring together, P(A) is the probability of event A happening, and P(B) is the probability of event B happening.
Conditional probability comes into play when events are not necessarily independent. It deals with the probability of one event occurring given that another event has already occurred. In the case of independent events, conditional probability simplifies greatly.
For independent events A and B, the conditional probability of event A occurring given that event B has occurred is simply:
In other words, if events A and B are independent, then the probability of A happening does not change based on the occurrence of B, and vice versa.
1. Independent Events:
a. Coin Tosses:
You are flipping a fair coin twice. What is the probability of getting heads on the first toss (Event A) and tails on the second toss (Event B)?
Since coin tosses are independent, we can calculate the probability of both events happening together as the product of their individual probabilities:
So, the probability of getting heads on the first toss and tails on the second toss is 0.25.
b. Dice Rolls:
You are rolling a fair six-sided die twice. What is the probability of rolling a 4 on the first roll (Event A) and a 6 on the second roll (Event B)?
Just like the previous example, since dice rolls are independent, we can calculate the probability of both events happening together as the product of their individual probabilities:
So, the probability of rolling a 4 on the first roll and a 6 on the second roll is
2. Dependent Events:
a. Drawing Cards:
You are drawing two cards from a standard deck of 52 cards without replacement. What is the probability of drawing a red card on the second draw (Event B) given that you drew a red card on the first draw (Event A)?
These events are dependent because the probability of the second event depends on the outcome of the first event. Let's calculate it step by step:
Probability of drawing a red card on the first draw (Event A):
After drawing a red card on the first draw, there are now 25 red cards left out of 51 cards for the second draw (Event B):
So, the probability of drawing a red card on the second draw given that you drew a red card on the first draw is
b. Weather Conditions:
Consider two weather events:
In this scenario, the probability of it raining today (Event A) might indeed depend on whether it rained yesterday (Event B). The idea here is that past weather conditions can influence current weather conditions. If it rained yesterday, the ground might be wet, and the atmosphere might still be conducive to rainfall, increasing the chances of rain today.
To illustrate this concept, let's assign some probabilities:
Probability of it raining today given that it didn't rain yesterday: P(_A_∣_B_′)=0.3 (This means there's a 30% chance of rain today if it didn't rain yesterday.)
Probability of it raining today given that it rained yesterday: P(A_∣_B)=0.6 (This means there's a 60% chance of rain today if it rained yesterday.)
Probability of it not raining today given that it didn't rain yesterday: P(_A_′∣_B_′)=0.7 (This means there's a 70% chance of no rain today if it didn't rain yesterday.)
Probability of it not raining today given that it rained yesterday: P(A_′∣_B)=0.4 (This means there's a 40% chance of no rain today if it rained yesterday.)
Now, you can see that these probabilities reflect the idea that past weather conditions influence the likelihood of current weather conditions. When it rained yesterday (Event B), there's a higher chance of rain today (Event A) compared to when it didn't rain yesterday (Event B').
This example illustrates dependent events in the context of weather conditions and how conditional probabilities help us understand the relationship between these events.
1. Independent Events:
a. You are rolling a fair six-sided die three times. Calculate the probability of getting a 6 on the first roll (Event A), a 3 on the second roll (Event B), and a 5 on the third roll (Event C). Are these events independent?
Rolling a fair six-sided die three times:
The probability of getting a 6 on the first roll (Event A), a 3 on the second roll (Event B), and a 5 on the third roll (Event C) can be calculated as:
These events are independent because the occurrence of one does not affect the occurrence of the others.
b. You are drawing two cards from a well-shuffled standard deck of 52 cards with replacement. Calculate the probability of drawing a red card on the first draw (Event A) and a black card on the second draw (Event B). Are these events independent?
Drawing two cards from a well-shuffled standard deck with replacement:
The probability of drawing a red card on the first draw (Event A) and a black card on the second draw (Event B) can be calculated as:
These events are independent because each draw is made with replacement, and the deck's composition doesn't change between draws.
2. Dependent Events:
a. You have a bag of 20 marbles, 8 of which are red, and 12 are blue. You draw one marble from the bag, record its color (Event A), and then draw a second marble from the same bag without replacement and record its color (Event B). Calculate the probability that both marbles are red (Event A and Event B).
b. Consider a situation where you are predicting the outcome of a soccer match (win, lose, or draw) for your favorite team (Event A) based on their performance history. Given that your team lost the last three matches (Event B), calculate the conditional probability that they will win the next match (P(A|B)).
To calculate the conditional probability that your favorite team will win the next match (Event A) given that they lost the last three matches (Event B), you can use the following formula:
Since the outcomes of soccer matches can be influenced by various factors, let's assume values for P(A_∩_B) and P(B) based on your knowledge or data. For example, if you estimate that your team wins 20% of the time (A), and they lost the last three matches (B), you can calculate:
To get the specific value of P(A_∣_B), you would need to provide the probability P(B) based on your assessment of your team's performance and the factors affecting the matches.
Bayes' Theorem is a fundamental concept in probability theory and statistics that provides a way to update the probability for a hypothesis as more evidence or information becomes available. It is named after the 18th-century British statistician and theologian Thomas Bayes, although the theorem itself was published posthumously in 1763 by his friend Richard Price.
The historical context of Bayes' Theorem is quite interesting. Thomas Bayes developed the theorem to address a problem related to inverse probability, which was a topic of great interest in the 18th century. Inverse probability problems involve determining the probability of a cause based on the observed effects, which is essentially what Bayes' Theorem accomplishes.
Bayes' Theorem remained relatively obscure until Richard Price rediscovered and published it. It gained significant attention because it provided a formal mathematical framework for updating beliefs or probabilities when new evidence is obtained. This was particularly important in fields like astronomy, where it could be used to refine predictions about celestial bodies' positions based on new observations.
Bayes' Theorem is of immense importance and applicability in various fields:
Bayes' Theorem, in its general form, is a mathematical formula used in probability theory and statistics to update the probability of a hypothesis or event based on new evidence. It is expressed as:
Components of the Formula:
Where Ai represents all possible hypotheses.
Role of Each Component in Bayesian Reasoning:
Bayes' Theorem is a fundamental tool for making decisions, predictions, and inferences in various fields, allowing for the incorporation of new data and evidence into existing knowledge or beliefs. It forms the basis of Bayesian statistics and reasoning.
Bayes' Theorem is applied in various real-world scenarios to make informed decisions, particularly when new evidence is available. Here are practical examples in different fields:
In all these applications, Bayes' Theorem plays a central role in updating probabilities based on new evidence. It allows for a systematic way to combine prior knowledge or beliefs with observed data to obtain more accurate and up-to-date probabilities. This updating process is particularly valuable when dealing with uncertainty and making decisions in dynamic and evolving situations. By incorporating new evidence, Bayes' Theorem helps individuals and organizations make more informed choices and predictions.
Bayesian inference is a broader statistical concept built upon Bayes' Theorem. While Bayes' Theorem provides a mathematical framework for updating probabilities based on new evidence, Bayesian inference is the entire process of using this framework to make probabilistic inferences, model data, estimate parameters, and make predictions. It encompasses a systematic approach to statistical reasoning that incorporates prior knowledge and evidence to draw conclusions.
Using Bayesian Inference for Statistical Modeling and Parameter Estimation:
Bayesian inference is particularly powerful for statistical modeling and parameter estimation because it allows practitioners to:
In essence, Bayesian inference provides a powerful and flexible framework for modeling complex systems, estimating parameters, and making predictions while accounting for uncertainty. It is widely employed in fields where probabilistic reasoning and the incorporation of prior knowledge are essential for making informed decisions.
Conditional probability and Bayes' Theorem are related concepts in probability theory, but they serve different purposes and address different aspects of probabilistic reasoning.
Definition: Conditional probability is the probability of one event occurring given that another event has already occurred. It answers the question: "What is the likelihood of event A happening, given that we know event B has occurred?"
Notation: It is denoted as P(A_∣_B), read as "the probability of A given B."
Formula: The formula for conditional probability is:
Here, P(A_∩_B) is the joint probability of both events A and B happening together, and P(B) is the probability of event B.
Use: Conditional probability is used to understand how one event's occurrence or knowledge about it affects the probability of another event. It's fundamental in scenarios like medical diagnosis, risk assessment, and many real-life decision-making processes.
Definition: Bayes' Theorem is a specific mathematical tool used for updating probabilities when new evidence becomes available. It answers the question: "What is the probability of a hypothesis or event A being true, given new evidence B?"
Notation: In the context of Bayes' Theorem, P(A) represents the prior probability of event A before considering evidence B, P(B_∣_A) is the likelihood of observing evidence B if A is true, P(B) is the probability of observing evidence B, and P(A_∣_B) is the posterior probability of A given B.
Formula: The formula for Bayes' Theorem is:
Use: Bayes' Theorem is employed when you want to update your beliefs or probabilities about a hypothesis or event in light of new evidence. It is commonly used in fields like statistics, machine learning, medical diagnosis, and decision analysis.
The key distinction is that conditional probability deals with the likelihood of events occurring given other events, without necessarily involving prior beliefs or updating. It helps answer questions like "What is the probability of rain given that it's cloudy?"
In contrast, Bayes' Theorem specifically addresses the updating of probabilities or beliefs based on new evidence. It is a tool for incorporating prior knowledge (prior probability) and new information (likelihood and evidence) to arrive at an updated probability (posterior probability) for an event or hypothesis. It helps answer questions like "What is the probability of a medical condition given a positive test result?"
While both concepts deal with probabilities and conditional relationships, Bayes' Theorem is a more versatile tool for Bayesian reasoning and updating probabilities in dynamic situations.
a. Accurate Prior Probabilities:
b. Data Availability and Quality:
c. Computational Complexity:
d. Model Misspecification:
e. Misconceptions and Misuses:
These advanced topics expand upon the foundational concepts of conditional probability and Bayes' Theorem, allowing for more sophisticated and powerful applications in various domains. However, they also come with their own complexities and challenges, such as model complexity and computational demands. Understanding these advanced topics can lead to more nuanced and accurate probabilistic modeling and inference.
Conditional probability and Bayes' Theorem are foundational concepts in probability theory and statistics. Conditional probability allows us to assess the likelihood of events given other events, while Bayes' Theorem provides a systematic way to update probabilities based on new evidence. These concepts find widespread applications in fields ranging from healthcare to finance, enabling more accurate predictions, informed decisions, and probabilistic reasoning.
1. Assume P(A)=0.2, P(B)=0.6,P(A U B)=0.5, Then P[A|B]=
Conditional Probability formula :- P (A | B) = P( A ∩ B ) / P ( B )
Inclusion exclusion principle :- P ( A U B ) = P( A ) + P( B ) – P( A ∩ B )
0.5 = 0.2 + 0.6 – P( A ∩ B )
P( A ∩ B ) = 0.3
Given Data :- P( A ) = 0.2 , P( B ) = 0.6
Put all the value in conditional probability formula
P (A | B) = 0.3 / 0.6 = 0.5
So option D is true
2. Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is :
The question is based on Bayes’ Theorem.
P(LED is Type 1) = 1/2
P(LED is type 2) = 1/2
Now, we need to see conditional probabilities.
P( LED lasting more than 100 hours / LED is Type 1) = 0.7
P( LED lasting more than 100 hours / LED is Type 2) = 0.4
3. Suppose a test is 99% accurate and 1% of people have a disease. What is the probability that you have the disease given that you tested positive? a. 1/2 b. 2/3 c. 1/4 d. 3/4
Let B be the event of testing positive and A be the event of having the disease.
We want to figure out P(A∣B). We know P(B∣A)=0.99 , which tells us we should use Bayes Theorem. Then, we also know P(B∣A’) = 0.01 and P(A)=0.01,P(A’)=0.99. We can plug this into the formula to get
Given that they passed the exam what is the probability it is a woman ?
5. Covid 19 has taken over the world and the use of Covid19 tests is still relevant to block the spread of the virus and protect our families.
You can follow the statistics of Covid 19 on the World Health Organization website: https://covid19.who.int/
If the Covid19 infection rate is 10% of the population, and thanks to the tests we have in Algeria, 95% of infected people are positive with 5% false positive.
What would be the probability that I am really infected if I test positive?
We will start multiplying the probability of infection (10%) by the probability of testing positive given that be infected (95%) then we divided by the sum of the probability of infection (10%) by the probability of testing positive given that be infected ( 95%) with not infected (90%) multiplied by false positive (5%)
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