Name: _________________________________________

This assignment will demonstrate the use of computer programs as experimental tools
for solving problems. Utilizing the `SequenceGenerator` class, you
will perform experiments with word distributions and attempt to estimate the
number of words with certain properties. Many of the assignments in this course will
similarly emphasize the use of computers for modeling complex phenomena or for
analyzing data, and so will help to develop empirical skills such as formulating
reasonable hypotheses, designing experiments, executing them, and critiquing the
results.

The SequenceGenerator class
contains a method named `randomSequence` that generates a random sequence of letters. The
method takes one parameter value, the length of the sequence to be generated, and returns a random
letter sequence.

EXERCISE 1:Load this class into the BlueJ IDE and create aSequenceGeneratorobject (with no parameter). Generate 10 random sequences of 3 letters and list them below. Were any of the 10 sequences real English words? Would you expect them to be?

EXERCISE 2:Similarly, generate 10 random sequences of 4 letters and list them below. Were any of the 10 sequences real English words? Would you expect them to be? Would you expect a 4-letter word be more or less likely than a 3-letter word? Explain.

EXERCISE 3:Now, create aSequenceGeneratorobject which utilizes an alphabet consisting of one letter, say "X". What would you expect to happen if you generated random letter sequences using this one-letter alphabet? Verify your predictions using the new object.

EXERCISE 4:Generating one sequence at a time can make experimentation with aSequenceGeneratortedious. Add the following method to the source code for theSequenceGeneratorclass, placed below the existing methods (but above the closing curly-brace)./** * Displays 5 random letter sequence of the specified length * @param seqLength the number of letters in the random sequences */ public void displaySequences(int seqLength) { System.out.println(randomSequence(seqLength) + " " + randomSequence(seqLength) + " " + randomSequence(seqLength) + " " + randomSequence(seqLength) + " " + randomSequence(seqLength)); } Compile the modified class, create a

SequenceGeneratorobject, and call your new method. Verify that this method behaves as described.

EXERCISE 5:Next, modify the newdisplaySequencesmethod by copy-and-pasting theSystem.out.printlnstatement 20 times inside the method. This should result in 100 random sequences being printed every time the method is called. Verify that your modified method behaves as described.

Using a `SequenceGenerator` object, you can perform some interesting
experiments. In particular, you can use the object as a tool for verifying or
disproving hypotheses about word distributions, and to generate further data for
analysis. First, consider the total number of unique 3-letter sequences that can
be generated. Since each of the three positions in a sequence can be any of the
26 letters, there are 26^{3} = 17,576 different
sequences. Clearly, not all of these sequences form real words. The question
arises: how many random 3-letter sequences would you expect to have to generate
before you obtain a word?

It so happens that there are approximately 500 3-letter words in the English
language (according to a popular online dictionary). Thus, if you generated a
random 3-letter sequence, there is a 500 out of 17,576 chance that it will be a
word. Since 500/17,567 is approximately 1/35, you might expect 1 out of every 35
sequences to be a word. More accurately, you might expect 28 out of every 1,000
random 3-letter sequences to be words, since 1/35 is approximately 2.8%. This
number can be verified experimentally using a `SequenceGenerator` object.

EXERCISE 6:Use aSequenceGeneratorobject to generate 1,000 random 3-letter sequences and count how many English words you obtain. List that number below.Hint:Generate the sequences in 10 groups of 100 by repeatedly callingdisplaySequencesand clearing the output screen after each call. Scanning 100 sequences for words can be done in just a few seconds.

Is the number you obtained close to the expected value of 28? If not, generate another 1,000 sequences, average the word counts, and see if the approximation improves.

As the length of the letter sequences increases, the chances of generating a
word at random decreases dramatically. For example, there are 1,777 4-letter
words in the online dictionary. Thus, the chances of generating a 4-letter word
at random are 1 in 257 (1,777/26^{4} =
1,777/456,976 = 1/257). For 5-letter sequences, the chances of generating a word
at random is 1 in 4,920 (2,415/26^{5} =
2,415/11,881,376 = 1/4,920) .

EXERCISE 7:Use aSequenceGeneratorobject to generate 1,000 random 4-letter sequences and count how many English words you obtain. List that number below.

Since there is only a 0.38% chance of generating a 4-letter word at random, you would expect to obtain around 4 words out of 1,000 random 4-letter sequences. Is the number you obtained close to 4? Compared to the case for 3-letter words, would you expect it to take more or fewer sequences to obtain a number close to the expected value? Explain.

Part of the blame for the scarcity of words among randomly generated sequences falls on letters such as 'q' and 'z'. Since these letters are used so infrequently in English, their inclusion in a random sequence of letters makes a real word extremely unlikely. If we exclude letters such as these, however, we can improve the chances of generating words considerably. For example, the 10 letters that appear most frequently in English text are "etaoinshrd". Random sequences of these letters would appear more likely to produce words.

EXERCISE 8:Create aSequenceGeneratorobject whose alphabet is limited to the letters "etaoinshrd". Generate 1,000 random 3-letter sequences of these letters and count how many English words you obtain. List that number below.Generate another 1,000 random letter sequences and count the number of English words. Is this number relatively consistent with the number obtained from the first 1,000 sequences (e.g., within 10% of each other)? If not, calculate the average of these two counts and perform the experiment again. Continue this process until the count obtained from the last 1,000 sequences is close (within 10%) of the average from the previous sequences. Show your results at each stage in the process.

Using your experimental results from the previous exercise, you should now be able to estimate the number of 3-letter words in the English language that use only the letters in "etaoinshrd". The followng general formula applies:

EXERCISE 9:Using the numbers you obtained in EXERCISE 8, estimate the number of 3-letter words in the English language that use only the letters in "etaoinshrd". Show your work in obtaining your estimate.

EXERCISE 10:Using the same approach as above, estimate the number of 4-letter words using only the letters "etaoinshrd". Show your work in obtaining your estimate.

Note:since there are many more 4-letter sequences than there are 3-letter sequences, it may take many more random generations in order to obtain a reasonable estimate. Justify your data.

Hand in a printout of your modifiedSequenceGeneratorclass along with these sheets.