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Now we return to Example 12, where n diners have checked their hats and we seek the probability that at least one diner is given his own hat at the end of the evening Let the events Ai denote the event diner i gets his own hat, so we seek P(A1 A2 An ) using the general addition theorem Suppose diner i gets his own hat There are (n 1)! ways for the remaining hats to be distributed, given the correct hat to diner i,so P(Ai ) = (n 1)!/n! There are n ways for a 1 single diner to be selected In a similar way, if diners i and j get their own hats, the remaining hats can be distributed in (n 2)! ways, so P(Ai Aj ) = (n 2)!/n! There are n ways for two diners to be chosen 2 Clearly, this argument can be continued We then nd that P(A1 A2 An ) = + which simpli es easily to P(A1 A2 An ) = 1 1 1 1 + 1! 2! 3! n! n (n 1)! 1 n! n (n 2)! 2 n! n (n n)! n n!

Figure 10.6 The layered structure of swsl-rules (SWSF, 2005).

Class CContactGroup public CContactItemPlusGroup De ned in cntitem.h This class is a concrete implementation of CContactItemPlusGroup that represents a contact card group. In addition to implementing virtual functions, it implements methods to manage the group label and access the list of included contact items. Public Methods static CContactGroup* NewL() static CContactGroup* NewLC() These methods create a new contact group. Contained Items Methods const CContactIdArray* ItemsContained() const This method returns a constant list of the items contained by the group.

n (n 3)! 3 n!

Table 24 shows some numerical results from this formula Table 24 n 1 2 3 4 5 6 7 8 9 p 100000 050000 066667 062500 063333 063194 063214 063212 063212

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seen as both speci cation and implementation language. SWSL-Rules language provides support for service-related tasks like discovery, contacting, policy speci cation, and so on. It is a layered-based languages as shown in Figure 10.6. The core of the SWSL-Rules language is represented by pure Horn sub-set of SWSL-Rules. This subset is extended by adding different features like (1) disjunction in the body and conjunction and implication in the head this extension is called monotonic Loyd-Topor (Mon LT) (Lloyd, 1987), (2) negation in the rule body interpreted as nation as failure this extension is called NAF. Furthermore, the Mon LT can be extended by adding quanti ers and implication in the rule body resulting in what is called nonmonotonic Loyd-Topor (Nonmon LT) extension. Other envisioned extensions are towards: (1) Courteous rules (Courteous) whit two new features: restricted classical negation and prioritized rules, (2) HiLog enables meta-programming, (3) Frames add object oriented features like frame syntax, types, and inheritance, (4) Rei cation allows rules to be referred and grouped. Finally, Equality can be possible extension as well. SWSL-FOL is a First-Order logic which includes features from HiLog and F-Logic. It has as well a layered structure which is depicted in Figure 10.7 Some of the extensions provided for SWSL-Rules apply for SWSLFOL as well. The only restriction is that the initial languages should have monotonic semantics. The resulting extensions depicted in Figure 10.7 are SWSL-FOL Equality, SWSL-FOL HiLog, and SWSL-FOL Frame.

CContactIdArray* ItemsContainedLC() const This method returns a list of the items contained by the group and leaves it on the cleanup stack. TBool ContainsItem(TContactItemId aContactId) This method checks whether a contact item is contained in the group. aContactId the contact item identi er to check. returns ETrue if the item is contained in the group. Group Label Methods TBool HasItemLabelField() This method tests whether the group has a label eld (of type KUidContactFieldTemplateLabel). TPtrC GetGroupLabelL() This method returns the group label (if any). void SetGroupLabelL(const TDesC& aLabel) This method sets the group label. aLabel the new group label to set.

It is perhaps surprising that, while the probabilities uctuate a bit, they appear to approach a limit To six decimal places, the probability that at least one diner gets his own hat is 0632121 for n 9 It can also be shown that this limit is 1 1/e for n 9

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