Suppose S={s1, s2, s3, s4, s5, s6}, P(s1)=P(s2)= P(s3)=1/4, P(s4)=p(s5)=1/16, P(s6)=1/8. What is the entropy of S? (Use log2)

The entropy \( H(S) \) of a discrete random variable \( S \) with probabilities \( P(s_i) \) is given by:

\[
H(S) = - \sum_{i=1}^{6} P(s_i) \log_2 P(s_i)
\]

Using the given probabilities for each element in the set \( S = \{s_1, s_2, s_3, s_4, s_5, s_6\} \):

\[
P(s_1) = P(s_2) = P(s_3) = \frac{1}{4}, \quad P(s_4) = P(s_5) = \frac{1}{16}, \quad P(s_6) = \frac{1}{8}
\]

Now, we calculate the entropy:

\[
H(S) = - \left( 3 \times \frac{1}{4} \log_2 \frac{1}{4} + 2 \times \frac{1}{16} \log_2 \frac{1}{16} + \frac{1}{8} \log_2 \frac{1}{8} \right)
\]

Each logarithmic term can be simplified:

\[
\log_2 \frac{1}{4} = -2, \quad \log_2 \frac{1}{16} = -4, \quad \log_2 \frac{1}{8} = -3
\]

Now substitute these values back into the equation:

\[
H(S) = - \left( 3 \times \frac{1}{4} \times (-2) + 2 \times \frac{1}{16} \times (-4) + \frac{1}{8} \times (-3) \right)
\]

Simplify each term:

\[
H(S) = - \left( 3 \times \frac{1}{4} \times (-2) + 2 \times \frac{1}{16} \times (-4) + \frac{1}{8} \times (-3) \right)
= - \left( -\frac{3}{2} + -\frac{1}{2} + -\frac{3}{8} \right)
\]

Combining terms:

\[
H(S) = \frac{3}{2} + \frac{1}{2} + \frac{3}{8} = 2 + 0.375 = 2.375
\]

Thus, the entropy of \( S \) is:

\[
H(S) = 2.375 \, \text{bits}
\]

Would you like further clarification on this or any specific step?

Here are 5 related questions:
1. How does entropy change with more equally probable outcomes?
2. How is entropy used in information theory?
3. What happens to entropy if one outcome becomes much more probable than others?
4. Can entropy ever be negative?
5. How do you calculate entropy when probabilities are not powers of 2?

**Tip:** Always double-check the probabilities sum to 1 to ensure correctness in entropy calculations.

Calculate the entropy of a discrete set S = {s1, s2, ..., s6} with probabilities P(s1) = P(s2) = P(s3) = 1/4, P(s4) = P(s5) = 1/16, and P(s6) = 1/8 using log base 2. This problem explores key concepts in information theory, such as entropy and probability. Suitable for advanced high school or undergraduate students.

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