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 $-\rm{log_2}\binom{52}{13} \approx 39.21 bits$.

 

[CellCoree]

The Shannon entropy problem in this scenario would involve determining the amount of information that can be conveyed through the nods and shakes of the player's head. In other words, how much uncertainty is reduced by the player's actions?

To find the Shannon entropy, we would first need to calculate the probability of each nod or shake of the head conveying a specific card. In this case, there are 52 cards in the deck and the player is trying to convey 13 of them, so the probability of each nod or shake representing a specific card would be 1/13.

Next, we would use the formula for Shannon entropy, H = -∑ p(x) log2p(x), where p(x) is the probability of each possible outcome (in this case, each nod or shake representing a specific card).

So, the Shannon entropy for this scenario would be:
H = -∑ (1/13) log2(1/13) = -13(1/13) log2(1/13) = -log2(1/13) = 3.7 bits

This means that the player's nods and shakes of the head can convey approximately 3.7 bits of information in this bridge game scenario. This measurement of entropy can help us understand the efficiency and effectiveness of the player's communication method and could be used to improve communication strategies in similar situations.