Questions tagged [bayes-theorem]

For questions related to Bayes' theorem, a result about conditional probabilities.

Bayes' theorem relates probabilities of events with conditional probabilities. In its most common form, the result states that if $A$ and $B$ are events, then

$$P(A | B) = \frac{P(B|A) P(A)}{P(B)}$$

where $P(A | B)$ is the conditional probability of $B$ given $A$.

Interpretations of this statement vary according to the Bayesian interpretation, and the Frequentist interpretation.

Reference: Bayes' Theorem.

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Is Bayes' Theorem really that interesting?

I have trouble understanding the massive importance that is afforded to Bayes' theorem in undergraduate courses in probability and popular science. From the purely mathematical point of view, I think it would be uncontroversial to say that Bayes'…
user78270
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Bayes' rule with 3 variables

I have been using Sebastian Thrun's course on AI and I have encountered a slightly difficult problem with probability theory. He poses the following statement: $$ P(R \mid H,S) = \frac{P(H \mid R,S) \; P(R \mid S)}{P(H \mid S)} $$ I understand he…
slyaer
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Bayes' Theorem with multiple random variables

I'm reviewing some notes regarding probability, and the section regarding Conditional Probability gives the following example: $P(X,Y|Z)=\frac{P(Z|X,Y)P(X,Y)}{P(Z)}=\frac{P(Y,Z|X)P(X)}{P(Z)}$ The middle expression is clearly just the application of…
Dongie Agnir
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Intuitive explanation for a constant answer in a Bayes theorem question

Question (previously asked here) You know there are 3 boys and an unknown number of girls in a nursery at a hospital. Then a woman gives birth a baby, but you do not know its gender, and it is placed in the nursery. Then a nurse comes in a picks up…
Gaurang Tandon
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Confusing probability question

I have got a task, which seems a quite confusing for me. It is simple: In a market, they sell eggs in egg holders, they store $10$ of them in each. There is $60$% chance, that all of the eggs are ok, $30$% chance, that exactly $1$ of them is broken,…
Atvin
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Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn't. We want to estimate $\mu$. Suppose further that the prior…
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Probability of getting heads given that first flip was a head?

What's the probability of getting heads on the second toss given that the first toss was a head. (Trying to refresh my probability a bit). I've seen this analyzed like this: HH 1/4 HT 1/4 TH 1/4 TT 1/4 So since we are given information (Head on…
Ole
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Why do knowers of Bayes's Theorem still commit the Base Rate Fallacy?

Source: 6 July 2014, "Do doctors understand test results?" by William Kremer, BBC World Service. A 50-year-old woman, with no prior symptoms of breast cancer, participates in routine mammography screening. She tests positive, is alarmed, and wants…
mr.slow
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Intuitively, why does Bayes' theorem work?

I'm not looking for a cryptic math demonstration. Rather, I'm interested in the intuition behind the theorem that reveals the a posteriori probability, given the prior probability $\times$ the likelihood.
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When to use Total Probability Rule and Bayes' Theorem.

Consider the following information about travelers on vacation: $40$% check work email, $30$% use a cell phone, $25$% bring a laptop with them, $23$% both check work email and use a cell phone, and $51$% neither check work email nor use a cell phone…
Kot
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What is the complement of conditional probabilities?

I am working with a problem that uses Bayes Theorem and conditional probabilities. I have the conditional probability that a plane has an emergency locator $(E)$ given that it was discovered $(D)$ which is $P(E\mid D)=0.60$. Now I am given that…
Kot
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A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

I have this question as an example in my maths school book. The solution given there is:- E = the man reports six P(S1)= Probability that six actually occurs = $\frac{1}{6}$ P(S2)= Probability that six doesn't occur= $\frac{5}{6}$ P(E|S1)=…
dknight
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Why do many textbooks on Bayes' Theorem include the frequency of the disease in examples on the reliability of medical tests?

A "standard" example of Bayes Theorem goes something like the following: In any given year, 1% of the population will get disease X. A particular test will detect the disease in 90% of individuals who have the disease but has a 5% false positive…
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Why do we refer to the denominator of Bayes' theorem as "marginal probability"?

Consider the following characterization of the Bayes' theorem: Bayes' Theorem Given some observed data $x$, the posterior probability that the paramater $\Theta$ has the value $\theta$ is $p(\theta \mid x) = p(x \mid \theta) p (\theta) / p(x)$,…
PP121
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Gambler coin problem: fair coin and two-headed coin

The following problem was posed to me: A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, i.e. the probability that the fair coin is selected is 0.5. When the gambler flips the chosen coin, it…
The Pointer
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