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A pair of independent desks

ProbabilityEasy

Desk A ends a day in one of $3$ states with probabilities $\frac{1}{2}$ , $\frac{1}{4}$ and $\frac{1}{4}$ . Desk B, independently, ends in one of $4$ equally likely states. What is the entropy of the pair of states in bits? Give a fraction.

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Hint

Entropies of independent variables add.

Solution

Desk A has $\frac{1}{2} \times 1 + \frac{1}{4} \times 2 + \frac{1}{4} \times 2 = \frac{3}{2}$ bits, and desk B has $log_{2} 4 = 2$ bits.
The pair has $\frac{3}{2} + 2 = \frac{7}{2}$ .

Answer: $\frac{7}{2}$

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