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.