How Much Information Can Nods and Shakes Convey in a Bridge Game?

[szpengchao]

consider a pack of 52 cards in a bridge game. a player try to convey 13 cards by nods of head or shake of heads to his partner. find the shannon entropy

 

[JSuarez]

You need $\rm{log}_{2}52 \approx 6 bits/card$ to specify a single card (admitting that they are all equiprobable). For 13 independent cards, you'll need $13\times\rm{log}_{2}52 bits$.

 

[szpengchao]

You need $\rm{log}_{2}52 \approx 6 bits/card$ to specify a single card (admitting that they are all equiprobable). For 13 independent cards, you'll need $13\times\rm{log}_{2}52 bits$.

but the question tells the answer is 40, and it asks to find a coding function with entropy 50

 

[JSuarez]

Well, then the question is asking for the amount of information necessary to transmit an arrangement of 13 cards as a whole and not individually; that was not clear from the question.

There are $\binom{52}{13}$ possible arrangements, and this gives an entropy of $-\rm{log_2}\binom{52}{13} \approx 39.21 bits$.