Consider the distribution of outcomes generated by N (N ≫ 1) independent throws of
(i) a coin or (ii) a six-sided dice. A coin (dice) is unbiased if both (all) its sides have equal
probability to show up in a throw; it is biased otherwise. For the cases (i) and (ii) above,
which of the following statements is/are true?
Options:
(A) The entropy of an unbiased coin is smaller than that of an unbiased dice.
(B) The entropy of an unbiased coin is greater than that of an unbiased dice.
(C) The entropy of a biased dice is smaller than that of an unbiased dice.
(D) The entropy of a biased coin is greater than that of an unbiased coin.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.53)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Nuclear & Particle Physics** within *Section 9: Nuclear and Particle Physics* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Nuclear & Particle Physics:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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