| After one more year Thorp published a book (I | | | | satisfactory. At this it is still worse if |
| mentioned it at the beginning of the article) | | | | the so called "order of substitution" is |
| in which he rather in details, in the form | | | | less, i.e. less is the number of these |
| comprehensible to any even a slightly | | | | movements (substitutions) after which the |
| literate and sensible person, set the rules | | | | cards are located in the same order they were |
| of formation of a winning strategy. But the | | | | from the start of a pack shuffling. In fact, |
| publication of the book did not only cause a | | | | if this number equals to t, then repeating |
| quick growth of those willing to enrich | | | | exactly similar movements any number of times |
| themselves at the cost of gambling houses' | | | | we, for all our wish, can not get more t |
| owners, as well as allowed the latter ones | | | | different positioning of cards in a pack, or, |
| to understand the main reason of | | | | using mathematical terms, not more t |
| effectiveness of the developed by Thorp | | | | different combinations of cards. |
| strategy. | | | | |
| | | | Certainly, in reality, shuffling of cards |
| First of all, casinos' owners understood at | | | | does not come down to recurrence of the same |
| last that it was necessary to introduce the | | | | movements. But even if we assume that a |
| following obligatory point into the rules of | | | | shuffling person (or an automatic device) |
| the game: cards are to be thoroughly shuffled | | | | makes casual movements at which there can |
| after each game! If this rule is rigorously | | | | appear with a certain probability all |
| observed, then a winning strategy of Thorp | | | | possible arrangements of cards in a pack at |
| cannot be applied, since the calculation of | | | | each single movement, the question of |
| probabilities of extracting one or another | | | | "quality" of such mixing turns out to be far |
| card from a pack was based on the knowledge | | | | from simple. This question is especially |
| of the fact that some cards would already not | | | | interesting from the practical point of view |
| appear in the game! | | | | that the majority of notorious crooked |
| | | | gamblers achieve phenomenal success using the |
| But what does it mean to have "thoroughly | | | | circumstance, that seemingly "careful |
| shuffled" cards? Usually in gambling houses | | | | shuffling" of cards actually is not such! |
| the process of "thoroughly shuffling" | | | | |
| presupposes the process when a croupier, one | | | | Mathematics helps to clear a situation with |
| of the gamblers or, that is still oftener | | | | regard to this issue as well. In the work |
| seen of late, a special automatic device | | | | "Gambling and Probability Theory" A.Reni |
| makes a certain number of more or less | | | | presents mathematical calculations allowing |
| monotonous movements with a pack (the number | | | | him to draw the following practical |
| of which varies from 10 to 20-25, as a rule). | | | | conclusion: " If all movements of a shuffling |
| Each of these movements changes the | | | | person are casual, so, basically, while |
| arrangement of cards in a pack. As | | | | shuffling a pack there can be any |
| mathematicians say, as a result of each | | | | substitution of cards, and if the number of |
| movement with cards a kind of "substitution" | | | | such movements is large enough, reasonably it |
| is made. But is it really so that as a result | | | | is possible to consider a pack "carefully |
| of such 10-25 movements a pack is thoroughly | | | | reshuffled". Analyzing these words, it is |
| shuffled, and in particular, if there are 52 | | | | possible to notice, that, firstly, the |
| cards in a pack then a probability of the | | | | conclusion about "quality" of shuffling has |
| fact that, for instance, an upper card will | | | | an essentially likelihood character |
| appear to be a queen will be equal to 1/13? | | | | ("reasonably"), and, secondly, that the |
| In other words, if we will, thus, for | | | | number of movements should be rather large |
| example, shuffle cards 130 times, then the | | | | (A.Reni prefers not to consider a question of |
| quality of our shuffling will turn out to be | | | | what is understood as "rather a large |
| more "thorough" if the number of times of | | | | number"). It is clear, however, that the |
| the queen's appearance on top out of these | | | | necessary number at least a sequence higher |
| 130 times will be closer to 10. | | | | than those 10-25 movements usually applied |
| | | | in a real game situation. Besides, it is not |
| Strictly mathematically it is possible to | | | | that simple "to test" movements of a |
| prove that in case our movements appear to be | | | | shuffling person (let alone the automatic |
| exactly similar (monotonous) then such a | | | | device) for "accidence"! |
| method of shuffling cards is not | | | | |