To understand why the entropy calculation in the image results in $0.571$, let's go through the calculation step-by-step.

Entropy Formula

Entropy $H$ in this context is calculated as: $H = - \sum_{i=1}^{k} P(\text{value}_i) \log_2 (P(\text{value}_i))$ where $P(\text{value}_i)$ represents the probability of each value $i$, and $k$ is the total number of distinct values.

Given Probabilities

In the provided example, there are four distinct probability values:

1. $P(\text{value}_1) = \frac{3}{8}$
2. $P(\text{value}_2) = \frac{2}{8} = \frac{1}{4}$
3. $P(\text{value}_3) = \frac{1}{8}$
4. $P(\text{value}_4) = \frac{2}{8} = \frac{1}{4}$

Step-by-Step Calculation

Using the entropy formula: $H = - \left( \frac{3}{8} \log_2 \frac{3}{8} + \frac{2}{8} \log_2 \frac{2}{8} + \frac{1}{8} \log_2 \frac{1}{8} + \frac{2}{8} \log_2 \frac{2}{8} \right)$

Now, calculate each term individually.

1. First Term: $\frac{3}{8} \log_2 \frac{3}{8}$ $\frac{3}{8} \approx 0.375$ $\log_2(0.375) \approx -1.415$ $\frac{3}{8} \times (-1.415) \approx -0.531$

2. Second Term: $\frac{2}{8} \log_2 \frac{2}{8}$ $\frac{2}{8} = 0.25$ $\log_2(0.25) = -2$ $\frac{2}{8} \times (-2) = -0.5$

3. Third Term: $\frac{1}{8} \log_2 \frac{1}{8}$ $\frac{1}{8} = 0.125$ $\log_2(0.125) = -3$ $\frac{1}{8} \times (-3) = -0.375$

4. Fourth Term: $\frac{2}{8} \log_2 \frac{2}{8}$ (same as the second term) $\frac{2}{8} \times (-2) = -0.5$

Summing the Terms

Now, sum all the terms: $H = -(-0.531 - 0.5 - 0.375 - 0.5)$ $H = 0.531 + 0.5 + 0.375 + 0.5 = 1.906$

It appears there was an inconsistency with the provided answer of $0.571$. The steps outlined above show the calculations in full detail based on the values given. If there is a need for further clarification on how $0.571$ was reached or if an error was present in the initial interpretation, I can re-evaluate with additional context.

Would you like further assistance, or do you have specific aspects you'd like to explore?

Here are 5 questions to expand on this explanation:

1. How does changing one of the probabilities affect the entropy calculation?
2. What is the significance of entropy in information theory?
3. How would you calculate entropy for a uniform distribution?
4. How does the base of the logarithm (e.g., $\log_2$, $\log_e$, $\log_{10}$) affect entropy?
5. Can entropy be negative, and if so, in what context?

Tip: Always double-check your logarithmic calculations for accuracy, especially with different bases.