What is Bayes theorem: Definition and 27 Discussions

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule; recently Bayes–Price theorem), named after the Reverend Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole.
One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in the theorem may have different probability interpretations. With Bayesian probability interpretation, the theorem expresses how a degree of belief, expressed as a probability, should rationally change to account for the availability of related evidence. Bayesian inference is fundamental to Bayesian statistics.

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  1. alima

    Comp Sci Which is the probability of JohnCalls given Burglary? Why?

    Questions: P (JohnCalls|Burglary) ? Why? Source of the image: Artificial Intelligence: A Modern Approach - Third Edition, by Stuart Russell and Peter Norvig. My attempt at solving: using Bayes' Theorem = P (A|B) = ( P(B|A) * P(A) ) / P(B) P(JohnCalls|Burglary) = P(J|B) = ( P(B|J) * P(J) /...
  2. jisbon

    Checking My Understanding of the Naive Bayes Theorem

    I would like to check my understanding here to see if it is correct as I am currently stuck at the moment. From the question, I can gather that: P(Rain | Dec) = 9/30 P(Cloudy | Rain) = 0.6? P(Cloudy | Rain) = 0.4 To answer the question: P(Rain | <Cloudy, Morning, December> ) = P(Rain) *...
  3. shivajikobardan

    MHB Understanding Bayes Theorem with an Example: 0.72 or 0.28?

    https://lh4.googleusercontent.com/FCqUErWAqlG8w0CskhcsLgpG91xyxzAkV_nD-bZAq8147-_RKesQDpglwqF5ylKZ0Q6VW88jX-KNuIpSXi9vhw5AiWmwiv_fMyyUo_WWZJG4uwWS0aB-3rGMA0h0PDo7ZpolexCe this is the question Here is a tutorial video but his steps are very confusing to me. I personally know bayes theorem and...
  4. M

    Bayes Theorem: coin flips and posterior predictive distribution

    Hi, I was attempting the following question and just wanted to check whether my working was correct: Question: A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the...
  5. K

    Why it doesn't sum to one in this simple naive Bayes classification?

    Summary:: When we have only three classes (Orange, Banana and Other) and three features (Long, Sweet and Yellow), why P(Other|Long, Sweet, Yellow) + P(Banana|Long, Sweet, Yellow) is not equal to 1 when P(Orange|Long, Sweet, Yellow) = 0 ? In this example...
  6. B

    Bayes' theorem and disease prevalence

    Hello at all! I have to solve this exercise: A tampon diagnostic test provides 1% positive results. The positive predictive values (probabilities of positive test disease) and negative (absence disease given negative test) are respectively 0.95 and 0.98. What is the prevalence of the disease...
  7. jim mcnamara

    Bayes Theorem Redux: Self-Correcting Probability of Correctness

    https://xkcd.com/2059 A new interesting self-referential theorem that "self-corrects" for the theorem's own probability of correctness. ?? Hold your pointer still in the middle if the cartoon for a few seconds to get P(C).
  8. Z

    Can Bayes Theorem Predict the Next Winner in a Team Matchup?

    Homework Statement Team 0 and Team 1 have played 1000 games and Team 0 has won 900 of them.[/B] When the two teams play next, knowing only this information, which team is more likely to win? Homework Equations P(X,Y) = P(YlX) x P(X) = P(XIY) x P(Y) (Not Sure) The Attempt at a Solution Hi, I...
  9. Tajeshwar

    What Is the Probability of Drawing a Specific Marble Color Sequence?

    Homework Statement An Urn contains two white marbles and one black marble. A marble is drawn from the Urn without replacement and put aside without my seeing it. Then a second marble is drawn, and it is white. What is the probability that the unknown removed marble is white, and what is the...
  10. Tajeshwar

    Bayes' Theorem: Probability of 8 Heads in 10 Tosses?

    Homework Statement Suppose I have a bent coin with a 60% probability of coming up heads. I throw the coin ten times and it comes up heads 8 times. What is the value of the “likelihood” term in Bayes’ Theorem -- the conditional probability of the data given the parameter. Homework...
  11. D

    Conditional probability reasoning problem

    Homework Statement Out of all the products a company makes 2% is damaged. During the routine control of the products, the products are put to a test which discovers the damaged ones in 99% of the cases. In 1% however it approves the damaged item as a working one and vice versa. Find the...
  12. R

    Bayesian Probability Distributions

    Hi, I was having some trouble doing some bayesian probability problems and was wondering if I could get any help. I think I was able to get the first two but am confused on the last. If someone could please check my work to make sure I am correct and help me on the last question that would be...
  13. Pushoam

    Probability of getting an ace in case of a loaded die

    Homework Statement Homework EquationsThe Attempt at a Solution 12)a) a) Since both events are independent : P( 33) = 9 / 441 b) Let’s have 4: throwing 4 Not 4 : not throwing 4 P ( 4 and not throwing 4 ) = P( 4) P ( not throwing 4) = (17*4)/441c) Let’s consider the sample...
  14. I

    Bayes theorem, answer way too small something wrong?

    im sure i followed it correctly but my answer is unusually small...1 in a thousand people have a disease. A company has discovered a new method for testing for the disease. If a person has the disease, the test will return a +ve result 99% of the time. If a person doesn't have the disease, the...
  15. A

    Using Bayes Theorem to Calculate Probability of a Student Being a Geek

    Homework Statement This problem was under applications of Bayes theorem, but I feel like I am bad at using it if that's the case: At a school 30% of the students are girls. 4% of the girls are geeks and 2% of all geeks are girls. What is probability that a random student is a geek...
  16. S

    When to use bayes theorem, and when to not.

    Just need the 'auto' portion answered. this is more of a general question. Homework Statement (This is problem #7 on the sample actuarial exam P, soa 153) An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders...
  17. I

    Dependent theorem and bayes theorem.

    Homework Statement One box of eggs has 12 eggs. The probability of all eggs being good is 70%. The probability of one being bad is 20%. The probability of two being bad is 10%. Given that 2 randomly selected eggs are good, what is the probability that all the eggs will be good? Homework...
  18. A

    Bayes Theorem for coins problem

    Homework Statement I have a bag of n coins, and 1 is fake - it has 2 heads. a) Determine the probability that if I flip a coin and it comes up heads, the coin is fake. b) If a coin is flipped k times and comes up heads k times, what is the probability that the coin is fake? Homework...
  19. P

    When to Use Total Probability Formula and Bayes Formula?

    Hello. I know that total probability formula is in Bayes formula/Theorem. But how do I know when I must use - total probability formula ? - Bayes formula? I want to know what is difference between total probability formula and Bayes formula/Theorem. I don't know when total...
  20. T

    Probability - using Bayes Theorem

    Homework Statement Question breaks down to this. defect occurs 1/100 items. .97 (97%) of the time when an item has a defect it is detected. .005 of the time, an item is detected to have a defect when it actually does not have one. What is the probability that an Actual defect occurs...
  21. S

    Multiple events and using bayes theorem

    I am not sure if I am doing this correctly, so please check my attempt: I have four discrete random variables: G, H, J, L Say, I want to find P(H|J, G, L). Then can I write as P(H|J, G, L) = P(J, G, L|H)*P(H) = P(J|G, L, H)*P(G|L, H)*P(L|H)*P(H) Are those equivalent? I can't find an...
  22. S

    Understanding the Difference Between P (A, B) and P (B, A)

    In proving Bayes' Theorem, we use the following two statements. P (A, B) = P (A|B) P (B) P (B, A) = P (B|A) P (A). I am wondering what's the difference between P (A, B) and P (B, A). Any takers?
  23. S

    What are the Odds of Having the Virus if the Test is Positive?

    Hey, A virus test is 98% accurate and 1 in 10000 people have the virus. Given that the test is positive, what are the chances that you have the virus? This is what I've got: Since keyword “given,” I’m assuming Bayes Theorem. So, let A be the event that you have the virus. Let B be the...
  24. L

    Understanding Bayes theorem for Probability

    1. At an electronics plant, there is an optional training program for new employees. From past experience it is known that 87% of new employees who attend the training program meet the production quota in the first week of work. It is also known that only 34% of workers who do not attend the...
  25. R

    Bayes Theorem Question - Am I doing this right?

    Another Conditional Problem... Lie Detector is 95% reliable when the person is guilty 98% reliable when innocent Random Person pulled from a pool of people... This pool is 6% guilty of theft and 94% have never stolen... Random person was determined guilty from the lie detector, what...
  26. H

    Bayes theorem and probability help

    Homework Statement 'A dashboard warning light is supposed to flash red if a car’s oil pressure is too low. On a certain model, the probability of the light flashing when it should is 0.99; 2% of the time, though, it flashes for no apparent reason. If there is a 10% chance that the oil...
  27. S

    Bayes Theorem probability of steroids

    Homework Statement When the test for steroids is given to soccer players, 98% of players taking steroids test positive and 0.5% of the players not taking steroids test positive. Suppose that 5% of soccer players take steroids, what is the probability that a player who tests positive takes...