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The chance the other child is a girl is 1 / 2 . This is a very different procedure from (1) picking a two-child family at random from all families with two children, at least one a boy, born on a Tuesday. The chance the family consists of a boy and a girl is 14 / 27 , about 0.52.
Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.
The corresponding Sylvester problem asks for 7 different S(2,3,9) systems of 12 triples each, together covering all () = triples. This solution was known to Bays (1917) which was found again from a different direction by Earl Kramer and Dale Mesner in a 1974 paper titled Intersections Among Steiner Systems (J Combinatorial Theory, Vol 16 pp 273 ...
Derangement. Permutation of the elements of a set in which no element appears in its original position. Number of possible permutations and derangements of a collection elements that number n. n! (n factorial) is the number of n -permutations; !n (n subfactorial) is the number of derangements – n -permutations where all of the n elements ...
The variant problem can be solved by the reflection method in a similar way to the original problem. The number of possible vote sequences is ( p + q q ) {\displaystyle {\tbinom {p+q}{q}}} . Call a sequence "bad" if the second candidate is ever ahead, and if the number of bad sequences can be enumerated then the number of "good" sequences can ...
A random permutation is a random ordering of a set of objects, that is, a permutation -valued random variable. The use of random permutations is common in games of chance and in randomized algorithms in coding theory, cryptography, and simulation. A good example of a random permutation is the fair shuffling of a standard deck of cards: this is ...
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information. [1][2] A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability ...
Problem of points. 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.