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As already remarked, most sources in the topic of probability, including many introductory probability textbooks, solve the problem by showing the conditional probabilities that the car is behind door 1 and door 2 are 1 / 3 and 2 / 3 (not 1 / 2 and 1 / 2 ) given that the contestant initially picks door 1 and the ...
A survey of psychology freshmen taking an introductory probability course was conducted to assess their solutions to the similar three-card problem. In the three-card problem, three cards are placed into a hat. One card is red on both sides, one is white on both sides, and one is white on one side and red on the other. If a card pulled from the ...
The problem is insolvable because any move changes by an even number. Since a move inverts two cups and each inversion changes W {\displaystyle W} by + 1 {\displaystyle +1} (if the cup was the right way up) or − 1 {\displaystyle -1} (otherwise), a move changes W {\displaystyle W} by the sum of two odd numbers, which is even, completing the proof.
The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
Each scenario has a 1 / 6 probability. The original three prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a 1 / 6 probability of occurring. However, the warden in the original case cannot reveal the fate of a pardoned prisoner.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
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 .
In the simplest case, if one allocates balls into bins (with =) sequentially one by one, and for each ball one chooses random bins at each step and then allocates the ball into the least loaded of the selected bins (ties broken arbitrarily), then with high probability the maximum load is: [8]