document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163933>Question 1163933</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please determine the number of bits to encode the part of speech of a sequence of words in English using a tag set of size 40. Assume the tag is a random variable. The following is the information that we have: \r<BR>\n" );
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document.write( "Among the 40 values of the tag, we know 90% of chances to be one of the four values:  ART (article), P (preposition), N(noun), and V(verb). These four values  are equally likely to be in a sequence. All other values of the tag has 10% chances which each of them is also equally likely to be in a sequence. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854367 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import math\r<BR>\n" );
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document.write( "p1 = 0.9 / 4<BR>\n" );
document.write( "p2 = 0.1 / 36\r<BR>\n" );
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document.write( "entropy = -(4 * p1 * math.log2(p1) + 36 * p2 * math.log2(p2))\r<BR>\n" );
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document.write( "print(f\"{p1=}\")<BR>\n" );
document.write( "print(f\"{p2=}\")<BR>\n" );
document.write( "print(f\"{entropy=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "p1=0.225<BR>\n" );
document.write( "p2=0.002777777777777778<BR>\n" );
document.write( "entropy=2.7859880937335126\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To determine the number of bits required to encode the part of speech of a word in a sequence, we calculate the **entropy** ($H$) of the tag set. In information theory, the entropy represents the average minimum number of bits needed to encode a random variable based on its probability distribution.\r<BR>\n" );
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document.write( "### 1. Identify the Probabilities<BR>\n" );
document.write( "The tag set has a total of 40 values, which are divided into two groups:\r<BR>\n" );
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document.write( "* **Group 1 (Frequent Tags):** There are 4 values (ART, P, N, and V). These account for 90% ($0.90$) of the occurrences and are equally likely.<BR>\n" );
document.write( "    * Probability of each tag ($P_{frequent}$) = $\frac{0.90}{4} = 0.225$<BR>\n" );
document.write( "* **Group 2 (Other Tags):** There are $40 - 4 = 36$ remaining values. These account for 10% ($0.10$) of the occurrences and are also equally likely.<BR>\n" );
document.write( "    * Probability of each tag ($P_{other}$) = $\frac{0.10}{36} \approx 0.002778$ (or $\frac{1}{360}$)\r<BR>\n" );
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document.write( "### 2. Calculate the Entropy ($H$)<BR>\n" );
document.write( "The formula for entropy is:<BR>\n" );
document.write( "$$H(X) = -\sum_{i=1}^{n} p_i \log_2(p_i)$$\r<BR>\n" );
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document.write( "Plugging in our values for the two groups:<BR>\n" );
document.write( "$$H(X) = - \left[ 4 \times (0.225 \log_2 0.225) + 36 \times \left( \frac{1}{360} \log_2 \frac{1}{360} \right) \right]$$\r<BR>\n" );
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document.write( "Breaking it down:<BR>\n" );
document.write( "* For the 4 frequent tags: $4 \times 0.225 \times \log_2(0.225) \approx 0.9 \times (-2.152) \approx -1.9368$<BR>\n" );
document.write( "* For the 36 other tags: $36 \times \frac{1}{360} \times \log_2(\frac{1}{360}) \approx 0.1 \times (-8.492) \approx -0.8492$\r<BR>\n" );
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document.write( "$$H(X) = -(-1.9368 - 0.8492) = 2.786 \text{ bits}$$\r<BR>\n" );
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document.write( "### Final Result<BR>\n" );
document.write( "The average number of bits required to encode the part of speech for each word in the sequence is approximately **2.786 bits**.\r<BR>\n" );
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document.write( "*(Note: If you were using a fixed-length encoding without considering probabilities, you would need $\lceil \log_2(40) \rceil = 6$ bits. However, based on the provided probability distribution, an optimal variable-length encoding like Huffman coding would achieve an average length close to the entropy of 2.786 bits.)* </SPAN>\n" );
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