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Bertrand's box paradox: the three equally probable outcomes after the first gold coin draw. The probability of drawing another gold coin from the same box is 0 in (a), and 1 in (b) and (c). Thus, the overall probability of drawing a gold coin in the second draw is 0 / 3 + 1 / 3 + 1 / 3 = 2 / 3 .
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand introduced it in his work Calcul des probabilités (1889) [1] as an example to show that the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite.
These questions ask for the probability of two different events, and thus can have different answers, even though both events are causally dependent on the coin landing heads. (This fact is even more obvious when one considers the complementary questions: "what is the probability that two red balls were placed in the box" and "what is the ...
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Non-probabilistic proofs were available earlier. Non-tangential boundary values [7] of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness. This result (Privalov's theorem), and several results of this kind, are deduced from martingale convergence. [8] Non-probabilistic proofs were available earlier.
In combinatorics, Bertrand's ballot problem is the question: "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count under the assumption that votes are counted in a randomly picked order?" The answer is
Notice that once we have fixed a probability distribution for X then the joint probability distribution of A, B is fixed, since A, B = X, Y or Y, X each with probability 1/2, independently of X, Y. The bad step 6 in the "always switching" argument led us to the finding E(B|A=a)>a for all a , and hence to the recommendation to switch, whether or ...
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory.One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.