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In rare cases, you may want to ensure that the class you create is never extended. The class might contain functionality that would be dangerous to modify. Another class might subclass yours, modify the behavior of one or more methods, and pass itself off as your class, since a subclass is also the type of its superclass. Of course, as the programmer, you ultimately have control over all the code going into your program, but when precompiled libraries are involved and projects become large, things can get out of control. Even if malicious code is not your concern, you can use this ability to signify that the class just should not be extended, that the class performs its sole duty, and there is no possible extra duty it can conceivably take on. It can be your signal that the class works best with composition instead of inheritance. For whatever reason you choose, you can make sure that nobody inherits your class by declaring it nal. To declare a nal class, simply add the final keyword to the class declaration.

In Example 3-8, a day s production of 850 manufactured parts contains 50 parts that do not conform to customer requirements. Two parts are selected at random, without replacement from the day s production. That is, selected units are not replaced before the next selection is made. Let A and B denote the events that the first and second parts are nonconforming, respectively. In 2, we found P1B A2 49 849 and P1A2 50 850. Consequently, knowledge that the first part is nonconforming suggests that it is less likely that the second part selected is nonconforming. This experiment is fundamentally different from the examples based on the binomial distribution. In this experiment, the trials are not independent. Note that, in the unusual case that each unit selected is replaced before the next selection, the trials are independent and there is a constant probability of a nonconforming part on each trial. Then, the number of nonconforming parts in the sample is a binomial random variable. Let X equal the number of nonconforming parts in the sample. Then P1X P1X 02 12 P1both parts conform2 1800 85021799 8492 0.886

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P1first part selected conforms and the second part selected does not, or the first part selected does not and the second part selected conforms) 1800 8502150 8492 150 85021800 8492 0.111 0.003 P1both parts do not conform2 150 8502149 8492

The final keyword can come either before or after the access control attribute (in this case public). By convention, it is placed after.

As in this example, samples are often selected without replacement. Although probabilities can be determined by the reasoning used in the example above, a general formula for computing probabilities when samples are selected without replacement is quite useful. The counting rules presented in Section 2-1.4, part of the CD material for 2, can be used to justify the formula given below. De nition A set of N objects contains K objects classified as successes N K objects classified as failures A sample of size n objects is selected randomly (without replacement) from the N objects, where K N and n N . Let the random variable X denote the number of successes in the sample. Then X is a hypergeometric random variable and K N K a ba b x n x N a b n

package { public final class Encrypter { public function encrypt(sourceText:String, key:String):String { //secure encryption algorithm goes here } } }

(3-13)

Now if someone passes you an Encrypter, you can be sure that it s actually this Encrypter, and not a subclass like this impostor:

The expression min 5K, n6 is used in the definition of the range of X because the maximum number of successes that can occur in the sample is the smaller of the sample size, n,

package { public class MaliciousEncrypter extends Encrypter { override public function encrypt(sourceText:String, key:String):String {

and the number of successes available, K. Also, if n K N, at least n K N successes must occur in the sample. Selected hypergeometric distributions are illustrated in Fig. 3-12. EXAMPLE 3-26 The example at the start of this section can be reanalyzed by using the general expression in the definition of a hypergeometric random variable. That is, a 50 800 ba b 0 2 850 a b 2 50 800 ba b 1 1 850 a b 2 50 800 ba b 2 0 850 a b 2

319600 360825

return sourceText; //mwa ha ha! } } }

40000 360825

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