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Jul
29
comment Probability of Union of Events in a Probability Product Space By Counting Event Size
@ zoli – It seems the extension theorem doesn’t provide answer to my question. However, from W. Feller’s Book, Vol I,ch v.4, I found that for series of independent experiments, instead of looking at the sample space as a whole we can consider it as a Cartesian product of individual spaces; allowing us to work with events/probabilities defined within each experiment. He somehow doesn’t speak about dependent events though. What I’m wondering, to start with, is – is it possible to construct a product space for series of dependent events – say Toss a coin and then roll a dice only if it’s a head ?
Jul
28
comment Union of three independent events
@Did - As far as efficient approach is concerned, are you talking about $$P(A_1 \cup A_2 \cup A_3) = 1 - P( (A_1 \cup A_2 \cup A_3)^c) = 1- P(A_1^c \cap A_2^c \cap A_3^c) = 1 - P(A_1^c).P(A_2^c).P(A_3^c)$$ Btw, not sure what do you mean by " First the formula for P(A∪B∪C) is known in full generality "
Jul
28
awarded  Commentator
Jul
28
comment Probability of Union of Events in a Probability Product Space By Counting Event Size
@ zoli - Well, you essentially saying that decomposing into experiments works iif experiments are pairwise independent and otherwise it won’t. It would be great if you have any logical/mathematical proof or some citation on this.
Jul
28
comment Probability of Union of Events in a Probability Product Space By Counting Event Size
My confusion remains unanswered. Let me rephrase it – For a sequence of experiments E1 and E2, irrespective of their dependency relation, can we solve probability question by defining events within each experiment? The solutions of option-2 (and -1), implicitly assume A and B are defined on same sample space. Whereas I would like to see if any alternate solution exists by decomposing the whole experiments and defining events within each experiment. Further I believe, one of the motivation for product space is to enable such decomposed experiments approach unless there is something I’m missing.
Jul
28
asked Probability of Union of Events in a Probability Product Space By Counting Event Size
Jul
27
answered Union of three independent events
Sep
23
awarded  Popular Question
Aug
5
comment Conditional probability - a formal discussion
@Did - I know that there exist quite a body of knowledge in this field and I must confess that I'm no expert in that. Having said that, I guess it would be appropriate to say the definition of the term 'conditional event' was mine even though I don’t think I coined the term myself. I rather believe I picked it from various papers (see Bibliography). Anyway, it seems you're knowledgeable in this field and that you're trying to say something. If so, please, enlighten me/us by all means. I would rather appreciate your candid (and hopefully elaborate) feedback, rather than some hinting comments.
Aug
5
comment Conditional probability - a formal discussion
Conditional Event: An event (A) when expressed under the knowledge of another event (H) is called a conditional event expressed as (A|H) <br/> Conditional Event Representation: Representing conditional events by using more formal symbols rather than using natural languages like English.
Aug
5
answered Conditional probability - a formal discussion
Jun
13
comment Why is the probability that a continuous random variable takes a specific value zero?
First, I think the relation should be $$ P(X = x) \leq \lim_{\epsilon \downarrow 0} [F(x) - F(x - \epsilon)] $$ and NOT $$ P(X = x) = \lim_{\epsilon \downarrow 0} [F(x) - F(x - \epsilon)]$$ Secondly, can we really justify taking limit on both side! Or should we say that the relation $$ P(X = x) \leq F(x) - F(x - \epsilon)$$ must hold even when $$\epsilon$$ is infinitesimally small i.e. $$ P(X = x) \leq \lim_{\epsilon \downarrow 0} [F(x) - F(x - \epsilon)] $$.
Jan
31
comment Maximization of probability using partial derivatives
Thanks André, you're right - we cannot use derivatives for maximization of discrete functions.
Jan
28
awarded  Supporter
Jan
27
asked Maximization of probability using partial derivatives
Dec
20
comment Cartesian Product Space with dependent event in probability!
Thanks Andre...am also in favor of 20 elements in $S_B$.
Dec
18
comment Order sequence of n numbers (with repetition) having 4 consecutive increasing numbers!!
@Hurkyl, Marc van Leeuwen: I thank you for your response and would like to know your views on this. Do you think this is a trivial question (that you or someone else can solve easily) or it's a challenging question (may be you would like to give a try as well given some spare time). P.S. (I'm neither a student nor this is any assignment from any course. I'm a software professional who takes interest in math.)
Dec
17
comment Order sequence of n numbers (with repetition) having 4 consecutive increasing numbers!!
@Hurkyl: I'm afraid that I could not guess whether you're criticizing the post or providing a hint to solution. In fact I didn't get why did you mention "no 4-sequence can begin with a digit in [N-2,N]"! To me if N $\ge $ 4+2=6 then a 4-sequence can start with N-2. Am I missing something!
Dec
17
asked Order sequence of n numbers (with repetition) having 4 consecutive increasing numbers!!
Dec
17
asked Cartesian Product Space with dependent event in probability!