Conditional probability on a finite set

[billschnieder]

T ≡ "two coins tossed 7 times by two people A and B giving outcomes $[A^+B^+, A^-B^+, A^+B^-, A^-B^+, A^+B^+, A^-B^-, A^-B^+]$, where + = heads and - = tails"
Calculate $P(A^+B^+|T)$, $P(A^+|T)$, $P(B^+|T)$ and $P(B^+|T,A^+)$

I asked this question elsewhere and there was a suggestion that the question does not make sense.

 

[mathman]

2/7, 3/7, 5/7, 2/3 (same order as question).

 

[harrylin]

2/7, 3/7, 5/7, 2/3 (same order as question).

Thanks for a first feedback; however, almost certainly the main issue is the title! The topic under discussion is about measurements to estimate the expectation values for an infinite sequence by the taking of many samples of a very large size. In view of that, the reaction to that example was:

Here you refer to the statistical analysis of a small sample. However, P usually stands for probability. As in your coin toss example, the probability of head (fair coin) P=0.5 even if you throw for example heads twice.
To elaborate, you could have the following statistical sequence:
++
++
While the statistical result of a few throws was ++ for all throws, this does certainly not mean that the probability of throwing ++ is 1.
- http://en.wikipedia.org/wiki/Law_of_large_numbers
- http://en.wikipedia.org/wiki/Student's_t-distribution

And one of the replies was:

Do you still believe that P(A^+B^+|T) is factorable into P(A^+|T) and P(B^+|T) [..]
What has small data set got to do with it? [the small finite set] "IS the population. Use your law of large numbers to randomly pick from that.

In view of the context of the discussion, the obvious follow-up questions are:

2. Is that the correct way to predict the outcome of a large sequence of coin tosses?
3. What can be said about factorizing the probabilities of the coins?

This kind of issue regularly pops up in discussions. Thus, thanks in advance for any clarifying comments by experts!

PS: The problem may stem in part from the different meanings that people attach to the word "probability", with even disagreeing schools of thought. For example, I define "probability" as in the introduction here:
http://en.wikipedia.org/wiki/Probability
Using that definition, your calculation is not a probability but a statistical result.

 

[billschnieder]

Thanks for a first feedback; however, almost certainly the main issue is the title!

Because it disagrees with your initial understanding of what you thought the question was about? Then you misunderstood the question.

The topic under discussion is about measurements to estimate the expectation values for an infinite sequence by the taking of many samples of a very large size.

Then you misunderstood the question. I never mentioned anything about an infinite sequence or taking samples. I gave you the set of outcomes which was finite.

2. Is that the correct way to predict the outcome of a large sequence of coin tosses?

This question is relevant only when a large sequence of coin tosses is being discussed. This is not and was not one of such a case. You incorrectly assumed it was and doubled down on that assumption despite my attempts to clarify the question to you.

PS: The problem may stem in part from the different meanings that people attach to the word "probability", with even disagreeing schools of thought. For example, I define "probability" as in the introduction here:
http://en.wikipedia.org/wiki/Probability
Using that definition, your calculation is not a probability but a statistical result.

Indeed I perceived that you thought that. It appears you still think the question as stated above does not make sense, or what do you mean by "is not a probability". Can you or can you not answer it as posed?

But I encourage you to check the first 4 chapters of "Probability Theory: The Logic of Science by ET Jaynes."

 

[harrylin]

[..] This question is relevant only when a large sequence of coin tosses is being discussed. This is not and was not one of such a case. You incorrectly assumed it was and doubled down on that assumption despite my attempts to clarify the question to you.

Can you clarify (for me as well as onlookers) what your question has to do in the context of measurements of as much data as is required for statistically valid estimations of expectation values?

[..] It appears you still think the question as stated above does not make sense, or what do you mean by "is not a probability". [..] But I encourage you to check the first 4 chapters of "Probability Theory: The Logic of Science by ET Jaynes."

I happen to have read those. Funny enough, I suspect that Jaynes would state that what you ask for is not probabilities but statements about a known statistical result. There is no degree of plausibility for those other than 1 or 0 when the even happened or not. Of course I could have overlooked or misunderstood something; if so, please present it.

 

[mathman]

Since P(... |T) (condition on T) was asked, the only thing that matters is the set of outcomes labelled T.

 

[harrylin]

Since P(... |T) (condition on T) was asked, the only thing that matters is the set of outcomes labelled T.

It depends on what you think may be meant with "P". Please explain what "P" means in your answer. Does it correspond to the likelihood that an event happened or will happen, or to the likelihood of unobserved things?

Note referring to jaynes, as both Bill and I appreciate his book: a probability is not the same thing as a frequency.

 

[mathman]

It depends on what you think may be meant with "P". Please explain what "P" means in your answer. Does it correspond to the likelihood that an event happened or will happen, or to the likelihood of unobserved things?

Note referring to jaynes, as both Bill and I appreciate his book: a probability is not the same thing as a frequency.

P to me simply means probability, a la Kolmogoroff axiom approach. For the example, the sample space has exactly 7 points and the random varables A and B assume values (+ or -) as given on those points.

 

[Stephen Tashi]

T ≡ "two coins tossed 7 times by two people A and B giving outcomes $[A^+B^+, A^-B^+, A^+B^-, A^-B^+, A^+B^+, A^-B^-, A^-B^+]$, where + = heads and - = tails"
Calculate $P(A^+B^+|T)$, $P(A^+|T)$, $P(B^+|T)$ and $P(B^+|T,A^+)$

I asked this question elsewhere and there was a suggestion that the question does not make sense.

It doesn't make sense because the notation is unclear.

For example, in the expression $P(A^+B^+)$ what is the meaning of the event $A^+B^+$? Does this mean the event that A and B both toss heads on the first of 7 tosses? Or does it mean that they both toss heads on at least one of 7 tosses? Or does it mean something else?

Furthermore, it doesn't make sense to speak of a conditional probability unless you have first established the probability space upon which you wish to place the condition. Since you didn't do that, people have to guess what the space is.

 

[harrylin]

P to me simply means probability, a la Kolmogoroff axiom approach. For the example, the sample space has exactly 7 points and the random varables A and B assume values (+ or -) as given on those points.

Well, the problem arose because I also think that P is meant to indicate probabilities or likelihoods. In contrast the provided data are supposedly factual outcomes (frequencies) - which should be distinguished from probabilities or likelihoods. Roughly speaking: statistical data are known facts while probabilities are bets. Assuming that we understood the information, the likelihood that A+B+ occurred is 1 - it's a sure bet. The frequency with which A+B+ occurred is 2/7, but that's not the same thing.

Based on that sample it may be possible to estimate for example an average expectation value with a certain probability, but I don't think that such was the question.

Bill can you elaborate on what you intended to show about probability calculations? Apparently you wanted to illustrate something about factorisation.

 

[Stephen Tashi]

Assuming that we understood the information, the likelihood that A+B+ occurred is 1 - it's a sure bet.

It might be a sure bet if A+B+ denoted an event. As I replied to billschneider, the notation A+B+ doesn't describe a specific event. It might mean the event "On a randomly selected toss from the 7 tosses, A and B both throw heads". The space of events we are considering hasn't been defined. It's billschneider that needs to be cross examined, not mathman.