In the book of Shiryaev, Probability, on page 26, it is given that

EXAMPLE 2. An urn contains M balls, m of which are "lucky." We ask for the probability that the second ball drawn is lucky (assuming that the result of the first draw is unknown, that a sample of size 2 is drawn without replacement, and that all outcomes are equally probable).Let A be the event that the first ball is lucky, B the event that the second is l lucky. Then [...] $$P(B)= m/M.$$ lt is interesting to observe that $P(A)$ is precisely $m/M$. Hence, when the nature of the first ball is unknown, it does not affect the probability that the second ball is lucky.

Even though I can verify the correctness of the derivation given for this result in the book, I think, there is still something missing;

Yes, since we have given no information about the result of the first drawn ball, one would expect that the probability of choosing a lucky ball should not be affected by its order. However, the experiment is done without replacement. This means compared to the first drawn ball, we have a piece of extra information about the state of the urn, namely that a ball has chosen and discarded afterwards, so we know that there are now $M-1$ balls inside the urn. But, the fact that $P(B) = m/M$ does not account for this, so how come ?


There are $M-1$ balls left. So what?

Draw all the balls and leave one remaining, without knowing any of the results. What is the chance that this remaining ball is lucky?

Now instead, draw all the balls and leave two remaining, without knowing any of the results.

What is the chance that the next draw is lucky?

Don't like balls? Think of a deck of cards, with some designated as lucky. See my answer here: Intuitive reason why sampling without replacement doesnt change expectation?.

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  • $\begingroup$ Oh, I see; thanks a lot. It is weird that there is no difference between knowing that some card is removed from the urn compared to not knowing it; it is like a piece of useless information. Do you know any theory that measures the uselessness of information in a probabilistic sense? $\endgroup$ – onurcanbkts Oct 27 '19 at 11:36

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