By Piotr Garbaczewski, Marek Wolf, Aleksander Weron

The research of chaotic behaviour of dynamical platforms has brought on new efforts to reconcile deterministic and stochastic tactics, in addition to classical and quantum physics. Efforts are being made to appreciate advanced and unpredictable behaviour. this article is an summary of those actions.

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**Extra resources for Chaos - The Interplay Between Stochastic and Deterministic Behaviour**

**Sample text**

I 20 ) of the numbers 1, . . , 20. The outcome (i 1 , i 2 , . . , i 20 ) corresponds to the situation in which i 1 is stamped in invisible ink on the outside of the first paper your friend chooses, i 2 on the second paper your friend chooses, etc. The total number of permutations of the integers 1, . . , 20 is 20 × 19 × . . × 1. The notation n! is used for the product 1 × 2 × . . × n (see the Appendix). Thus, the sample space consists of 20! 1 different elements. Each element is assigned the same probability 20!

A player who plays this game a large number of times reasons intuitively as follows in order to determine the average win per game in n games. 05n repetitions of the game, the player wins 2 and 3 dollars, respectively. 05 dollars (meaning that the average “win” is actually a loss). 05 is said to be the expected value of X . The expected value of X is written as E(X ). In the casino game E(X ) is given by E(X ) = (−1) × P(X = −1) + 2 × P(X = 2) + 3 × P(X = 3). 3 Expected value and the law of large numbers 33 The general definition of expected value is reasoned out in the example above.

The public chooses four candidates for the final round. Each candidate has an equal probability of being chosen. 5% of getting through to the final: they give her a 12 probability 1 1 of being chosen first, a 11 probability of being chosen second, a 10 probability of being chosen third, and a 19 probability of being chosen fourth. Is this calculation correct? 3 A dog has a litter of four puppies. Set up a probability model to answer the following question. Can we correctly say that the litter more likely consists of three puppies of one gender and one of the other than that it consists of two puppies of each gender?