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