Questions tagged [bayes-theorem]
For questions related to Bayes' theorem, a result about conditional probabilities.
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Bayes' Theorem Problem
I am having trouble trying to solve this problem. Anything helps!
Question: When doing a lab, the professor instructs students to get some rocks from a bucket and study them under a microscope. He ...
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Bayes' Theorem About Drawers and T-Shirts
Drawer 1 has 4 black t-shirts. Drawer 2 has 4 white t-shirts. Drawer 3 has 3 black t-shirts and 1 white t-shirt. Assuming that I drew a black t-shirt from a random drawer, what is the probability that ...
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Help a misunderstood Bayesian mathematician
I'm a mathematician-turned-physician and I'm currently taking a Genetics course. Some high-school level probability is usually covered in these courses and it would be expected that lecturers that are ...
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How is Bayes' theorm different from the theorem of conditional probability?
The theorem of conditional probability states that
$$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$$
Bayes' theorem on the other hand tells us the probability that a prior condition is true if a given event ...
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Probabilty question on correct option
I was asked this question in an interview of data science and I found it hard to find the right approach . Kindly help .
Below is the question:
Check the following statements from 1-3 , each statement ...
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Law of Total Probability and Commas
I have a question regarding the use of commas when writing probabilities. In the following statement of the law of total probability for conditional probabilities, commas are used to describe unions, ...
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Linear combination of posterior g.p.d.f
Suppose that each of $k$ statisticians has his own prior distribution for a certain
parameter $\theta$, and let $f_{i}$ be the g.p.d.f. which statistician $i$ assigns to $\theta$, $i = 1, ..., k$.
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Why am I getting an incorrect answer from Bayes probability rule?
The original question follows:
If a [fair] coin is flipped 16 times, what is the probability of landing exactly 5 heads given that at least 3 of the flips landed heads?
Approach 1- Bayes Rule
My ...
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Bayesian Theory's likelihood meaning and model
I am trying to understand Bayes' Theorem when applied in Machine Learning.
Basically, Bayes' Theorem is given by the formula:
$$ P(A\&B|B) = \frac{P(B\&A|A)*P(A)}{P(B)} $$
I understand how the ...
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Can any posterior follow given an appropriate prior?
Let's assume we are given some observation $x$, and two distributions $p(x | \theta)$ and $p(\theta | x)$. What are the necessary conditions that allow a prior $p(\theta)$ to exist, such that Bayes' ...
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How would you calculate the posterior distribution?
Suppose that you have a stocastic variable:
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$$ Y\mid\Theta\sim Laplace\left(\mu,\sigma^2\right) $$
(2)
$$ \Theta\sim N\left(\mu_0,\sigma^2_0\right) $$
and
(3)
$$ Z\sim Laplace\left(\mu,\sigma^2\...
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Is this probability statement correct?
If P(a|b) = P(a), then P(a|b,c) = P(a|c)
I think this is a correct statement. If a and b are independent, then it makes sense that the probability of a given b and c is the same as the probability of ...
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Bayes theorem with medical tests
Suppose after a set of tests you are 75% sure you have the antigen in your body. Then you run one more test. It returns positive. Now you are 95% sure. The false positive rate is 10%. What can you ...
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Confirming my understanding of posterior, marginal, and conditional distributions
I'm learning Bayesian statistics and want to verify my understanding of a few things.
Let's say my data $X$ follows the model $f(x \mid \theta)$ with a prior $\pi(\theta)$. After observations $\...
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How data goes to the other side in maximum likelihood estimation
I am reading the book Mathematics for Machine learning and I am a bit confused with maximum likelihood estimation. I understand that the likelihood is the probability of get certain observations $x$, ...
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How many degrees of freedom are in a Bayes type scenario?
Given a real world condition (such as a medical condition) which may or may not be present, and an indicator (such as a medical test) predicting whether or not that condition is present, there are ...
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Is proportionality in log space valid? (Example: Bayes Theorem)
For Bayes we can write
$$
p(X|Y) = \frac{p(X)\ p(Y|X)}{p(Y)} \propto p(X)\ p(Y|X)
$$
in log space we can use the sum, for example when we want to calculate the maximum.
$$
argmax_k\ p(X=k|Y) = ...
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What exactly is the difference between the Theorem of total probability and Baye's theorem?
The theory of total probability for dependent events states that
$ P(A) = \sum\limits_{i=1}^nP(A|E_i)P(E_i)$ and $P(A|E_i) = \frac{P(A \cap E_i)}{P(E_i)}$
which in my eyes is the same as saying $ P(...
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Bayesian posterior probability and conditional probability
This is a homework exercise but I am stuck, and I believe there is something basic that is still confusing me. This is the problem:
There is a test for a new illness. The lab who developed did the ...
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Help with Bayes theorem question
I am trying to understand how to apply Bayes Theorem to solve the following question:
A drug manufacturer claims that its roadside drug test will detect the presence of cannabis in the blood 90% of ...
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Why is the numerator of Baye's Theorem $P(A\cap B)$ instead of $P(A|B)$?
For example, say I have am holding an object; $O$ is the event that the object is an orange, and $R$ is the event that the object is round.
$P(O|R) = \frac{P(R|O)}{P(R)}$
This is obviously incorrect, ...
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Bayesian statistics (Finding a posterior distribution)
Assume that when $\theta=1$, Y ~ N(1,$\sigma^2$) and when $\theta=2$, Y ~ N(2,$\sigma^2$). Let P($\theta=1$)=P($\theta=2$)=0.5
I'd like to find the posterior distribution. Here's the step I've done.
(...
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Can someone please verify my solutions for this probability question on bayes' theorem?
Assume a COVID test can identify the presence of COVID, given that the person has COVID, with probability $p_d$. Assume the test assigns false positives with probability $p_f$ (a test will be positive ...
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Can we take out this term out of the integrand?
Suppose we have three (say continuous) variables $X$, $Y$ and $Z$. We know that may express the conditional probability $P[x\mid y]$, as follows:
\begin{equation}
P[x\mid y]=\int_{\mathbb{R}} P[x,z\...
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You have a fair coin and a biased coin. What is the probability the second flip will be heads if the first flip was?
Derived from example 2.5.10 from Blitzstein and Hwang (page 65).
Assume $P( heads | fair )=1/2$ and $P( heads | biased )=1/4$.
If we let F be the event we draw the fair coin, $A_1$ be the event the ...
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Valid circumstances where P(A) = P(A|B)⋅P(B)
Normally we would say that:
$$\ P(A\cap B) = P(A|B)⋅P(B) = P(B|A)⋅P(A)
$$
but I am wondering if there are any valid circumstances where this would hold true:
$$\ P(A) = P(A|B)⋅P(B) $$
I was thinking ...
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Example of Bayes Theorem...
In the paternity suit, the mother's blood type is A, the man's blood type is B, and the child's blood type is AB. According to various circumstances, the possibility that the man identified as a real ...
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Polya's Urn solution to question
A Polya urn has two balls, one red, and one blue. One of these is chosen
uniformly at random. It is put back, with another of the same color. Again, a ball is chosen
uniformly at random, and put back, ...
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Bayes Theorem Problem
An insurance company classifies customers as accident-prone or not accident-prone. An accident-prone customer has a 0.3 probability of submitting a claim each year. A
customer who is not accident-...
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Bayes Theorem Assumptions [duplicate]
A man is known to speak the truth 3 out of 5 times. He throws a die, and reports that it is a 1. Find the probability that it is actually 1. (CBSE 2014)
Solution 1
Assume the die is fair.
If the man ...
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Need help with a math problem (Solvable with Bayes Theorem?) Also involves entropy
The Question goes as follows:
Research has proven that 70% of the men has dark hair and that 25% of the
women is blond. Furthermore it is known that 80% of the blonde women marries a dark-haired
man.
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Conditional Probability Decomposition of n>=3 elements into {1,2} elements
Is it possible to decompose a conditional probability with three or more elements (i.e. number of events $n>=3$, where $n$ is the number of elements or events) into conditional probabilities of ...
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Expectations of probabilities and weighted coins
Came across an interesting question recently and didn't see an answer on here already:
A casino produces coins which, when flipped, will land on heads with probability $p$. The coins are weighted, ...
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Could you please solve this question on Bayes theorem without applying bayes theorem?
The contents of 3 urns are: 1 white, 2 red, 3 green balls; 2 white, 1 red, 1 green balls and 4 white, 5 red, 3 green balls. Two balls are drawn at random. These are found to be one white and one green....
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Probability of a white ball chosen from a bag if there are two bags.
I was reading Bayes' Theorem and I came across a question. The question was:
What is the probability of picking a white ball from a bag, when there are two bags, Bag A and Bag B?
Bag A contains $2$ ...
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Likelihood of knowing the answer given y correct answers on multiple choice test
I've been trying to solve the following problem but am thrown off by the "guessing in case she doesn't know the answer". The problem is as follows:
A student takes a test with n questions ...
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Why is this coin-flip not memoryless and independent of past coin-flips for a fair coin, so we have to use bayes theorem and conditional probability?
A man flips a fair coin with sides heads and tails five times. Given that the man receives heads on at least two of the coin flips, what is the probability that he receives tails exactly twice after ...
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How can I derive the mathematical relationship between P(A|B,C) with P(A,B), P(A,C) and, P(B,C) if any.
I am trying to model a process mathematically, where I have three events. $(X=x)$, $(U=1)$, and $(T=t)$. I am interested in finding $\mathbb{P}(U=1\mid X=x,T=t)$.
From my model, I know that the event $...
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Updating the covariance of bivariate normal after a signal
Let $X$ and $Y$ be two bivariate normally distributed random variables with means $\mu_X$ and $\mu_Y$ and variances $\sigma_X^2$ and $\sigma_Y^2$. The covariance is $Cov(X,Y) = \rho\sigma_X\sigma_Y$, ...
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Difficulties in understanding a conditional probability question from a textbook
In the book "Complete Probability & Statistics 1 for Cambridge International AS & A Level" I found the following question:
The homework diaries and completed homework of two ...
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Bayesian Inference Multiplication
I have been reading chapter two of the book Statistical Rethinking: A Bayesian Course with Examples in R and Stan and I can't see why for the marble example, successive multiplications will produce ...
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Conditional expectation with multiple random variables
I'm trying to understand the solution to a problem from A First Course in Probability (Ross):
There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped ...
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Probability that a car accident is correctly attributed to faulty brakes
The probability that a one-car accident is due to faulty brakes is 0.04, the probability that a one-car accident is correctly attributed to faulty brakes is 0.82, and the probability that a one-car ...
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Bayes' Theorem in Conditional Probability
The scenario given by the problem is as follows:
We are testing for a disease D that we think is present, D+, with probability 0.4, and absent, D-, with probability 0.6. We believe that a test has ...
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Applying Bayes rule for measure-zero events
Suppose we have a set of states $\Theta$ and a set of messages $M$. The prior measure is $\mu_0 \in \Delta \Theta$.
A signal $\sigma: \Theta \times M \to [0,1]$ is a regular conditional probability (...
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Coin tossing game - probabilities of $(K,K,K)$ and $(Z,K,Z)$
Let's consider the following game of two players:
A fair coin is tossed as long as no one has won. Player $A$ wins if the sequence $(Z,K,Z)$ appears and player $B$ wins if $(K,K,K)$ appears. What is ...
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Denominator intractability in Bayesian inference
It can often be read that Bayesian inference i.e. using Bayes theorem, to compute the posterior i.e.:
$p(\theta|x)$, with $\theta$ a continuous random variable (and possibly $x$ too)
can be ...
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Conditional probability with 4 coins
The question: A box contains four coins, two of which are fair, one double-headed (i.e., heads on both sides), and the third is biased in such a way that it comes up heads with probability 1/4. A coin ...
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What is the probability a person is infected given two independent tests are positive
I know this question has been asked a couple of times, but I'm not sure that I understood the calculation properly. I wanted to get some clarity and make sure I'm doing the calculation correctly. If I ...
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Does Bayesian Theorem mean that testing of asymptopmatic people has more probability of False positives?
https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_probability/bs704_probability6.html
A patient goes to see a doctor. The doctor performs a test with 99
percent reliability--that is, 99 percent of ...
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