How do we post a comment on a problem and its answer? I haven't seen anywhere that you can click to make a comment, ask a question, or give an answer, even after having signed up and PAID.

This is a ridiculous statement: "Ensure this value has at most 100 characters (it has 368)." If I want to write 500, or 1000, then so what?

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`*italics*`

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boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

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## Comments

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TopNewestHi Dale,

For a select few of our problems, we have locked the solution discussions to keep them clean. The problem Marbles with an equal chance is one of them.

With regards to MCQ options being less than 100 characters, that is a restriction placed to ensure that the options are reasonably visible. You will not be allowed to have more than 100 characters in the option. If you have a really long option choice, you can state the options in the problem statement, and then have answer options of "A, B, C, D".

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The axioms of probability must be obeyed. If you work out a set of the probabilities of events that are mutually exclusive and mutually exhaustive, these probabilities must add up to unity. All probabilities must be nonnegative. If two events, A and B, are disjoint, then Pr(A or B) must equal Pr(A) + Pr(B). Simple checks like these can uncover whopping mistakes, and they can also give you confidence in your answers.

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Are you trying to say that the answer to "Marbles with an equal chance" is incorrect?

If so, can you explain your reasoning? If you look at Pulkit Gupta's solution, which aspect do you disagree with?

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No, I am just telling you that WRITING OUT a "sanity check" on the solution to any probability problem is good sense. Why is it that YOU find this concept (or requirement) so hard to understand? I can understand it somewhat from the responses of some of my students. They were just dumb and unable to comprehend. Once of my students told me this. When I wrote "Check your answer" ON THE TEST PAGE, he thought I meant "to just look at it really hard". Of course, I had absolutely no evidence that anyone had done so or not, because nobody wrote anything down.

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I wanted written sanity check of some sort - that I could see and read. One of my colleagues told be that I should have docked all of them some points for not writing anything down when a had instructed "Check your answer." If you have a problem with three mutually exclusive and mutually exhaustive events, with probabilities 1/2, 1/3, and 1/6, you just write down 1/2 + 1/3 + 1/6 = 6/6 = 1, and that suffices as a check on the sanity of the answer.

If you got 1/2, 1/3, and - 1/6, the instant sanity check is - 1/6 < 0, YIKES!

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Ah ic. Thanks for explaining.

I was under the impression that you disagreed with the answer, which is why I asked "are you trying to say that the answer is incorrect?". You then clarified that you running a sanity check is a good way to correct for errors.

I agree with you that double checking the validity of the answer would help those who already have a strong foundation spot their mistakes. Unfortunately, most of those who answer incorrectly (and don't understand why) fall into the camp of not understanding the theory, and would "just look at it really hard".

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I wished to make a response that was completely verbal, and so whether it was 100, 200, 368, or 500 characters long makes no difference as far as legibility is concerned. I just wanted to point out that in probability problems like this, it is invaluable to make a formal written check on the answer to make sure it makes sense.

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Other (usually) simple sanity checks on probability problems are that if events A and B are independent, then you must have Pr(A and B) = Pr(A)Pr(B). Also, you need to verify that all probability density functions f(x) are always nonnegative.

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Some respondents on some other problems have pointed out that all integrals that you use must converge and that all infinite summations must converge also. In some problems, such as when using the Cauchy probability density, you can get away with using the Cauchy principal value (CPV) of integrals if you are careful and you can justify that use. Often, the use of the CPV can be justified visually from the graph of the probability density function.

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