Recent questions in Survey Questions

Recent questions in Survey Questions
Survey Questions Answered
Melina Barber 2022-09-19

I recently have this question:
I have a bag of toys. 10% of the toys are balls. 10% of the toys are blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?
One person provided an answer immediately and others suggested that more details were required before an answer could even be considered. But, there was a reason I asked this question the way that I did.
I was thinking about probabilities and I was coming up with a way to ask a more complicated question on math.stackexchange.com. I needed a basic example so I came up with the toys problem I posted here.
I wanted to run it by a friend of mine and I started by asking the above question the same way. When I thought of the problem, it seemed very clear to me that the question was "what is P ( b l u e b a l l )." I thought the calculation was generally accepted to be
P ( b l u e b a l l ) = P ( b l u e ) P ( b a l l )
When I asked my friend, he said, "it's impossible to know without more information." I was baffled because I thought this is what one would call "a priori probability."
I remember taking statistics tests in high school with questions like "if you roll two dice, what're the odds of rolling a 7," "what is the probability of flipping a coin 3 times and getting three heads," or "if you discard one card from the top of the deck, what is the probability that the next card is an ace?"
Then, I met math.stackexchange.com and found that people tend to talk about "fair dice," "fair coins," and "standard decks." I always thought that was pedantic so I tested my theory with the question above and it appears you really need to specify that "the toys are randomly painted blue."
It's clear now that I don't know how to ask a question about probability.
Why do you need to specify that a coin is fair?
Why would a problem like this be "unsolvable?"
If this isn't an example of a priori probability, can you give one or explain why?
Why doesn't the Principle of Indifference allow you to assume that the toys were randomly painted blue?
Why is it that on math tests, you don't have to specify that the coin is fair or ideal but in real life you do?
Why doesn't anybody at the craps table ask, "are these dice fair?"
If this were a casino game that paid out 100 to 1, would you play?
This comment has continued being relevant so I'll put it in the post:
Here's a probability question I found online on a math education site: "A city survey found that 47% of teenagers have a part time job. The same survey found that 78% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part time job and plans to attend college?" If that was on your test, would you answer "none of the above" because you know the coincident rate between part time job holders and kids with college aspirations is probably not negligible or would you answer, "about 37%?"

Survey Questions Answered
Pranav Ward 2022-09-13

I am studying a Tutorial on Maximum Likelihood Estimation in Linear Regression and I have a question.
When we have more than one regressor (a.k.a. multiple linear regression1), the model comes in its matrix form y = X β + ϵ, (1)where y is the response vector, X is the design matrix with each its row specifying under what design or conditions the corresponding response is observed (hence the name), β is the vector of regression coefficients, and ϵ is the residual vector distributing as a zero-mean multivariable Gaussian with a diagonal covariance matrix N ( 0 , σ 2 I N ), where I N is the N × N identity matrix. Therefore y N ( X β , σ 2 I N ), (2)meaning that linear combination X β explains (or predicts) response y with uncertainty characterized by a variance of σ 2 .
Assume y, β, and ϵ R n Under the model assumptions, we aim to estimate the unknown parameters ( β and σ 2 ) from the data available (X and y).
Maximum likelihood (ML) estimation is the most common estimator. We maximize the log-likelihood w.r.t. β and σ 2 L ( β , σ 2 | y , X ) = N 2 log 2 π N 2 l o g σ 2 1 2 σ 2 ( y X β ) T ( y X β )
I am trying to understand that how the log-likelihood, L ( β , σ 2 | y , X ), is formed. Normally, I saw these problems when we have x i as vector of size d(d is number of parameter for each data). specifically, when xi is a vector, I wrote is as
ln i = 1 N 1 ( 2 π ) d σ 2 exp ( 1 2 σ 2 ( x i μ ) T ( x i μ ) ) = i ln 1 ( 2 π ) d σ 2 exp ( 1 2 σ 2 ( x i μ ) T ( x i μ ) ) . But in the case that is shown in this tutorial, there is no index I to apply summation.