Question

A computer key board has 108 keys. Assuming equal probability of any key pressing

calculate the information content, I, carried by each key. Prove that this information

is more than the information carried by a key on the number pad of a touch tone

telephone. [Hint: Touch phone has 16 keys on its pad]

Answer 1

As per the given question,

Total number of available key(n) =108

Probability to press any one key $P(x)=\dfrac{1}{108}$

As per the Shanon entropy theorem,

$=-\Sigma_n P(x)\log(\dfrac{1}{P(x)})$

$=-\Sigma_n 108\log(\dfrac{1}{1/108})$

$=-108\times \dfrac{1}{108}\log{(108)}$

$=-\log(108)$

Similarly, For touch pad,

Total number of key (n)=16

Probability of press any one key $P(x)=\dfrac{1}{16}$

$=-\Sigma_n P(x)\log(\dfrac{1}{1/P(x)})$

$=-16\times\dfrac{1}{16}\log{\dfrac{1}{(1/16)}}$

$=-\log(16)$

Hence, from the above it is clear that this information is more than the information carried by a key on the number pad of a touch tone telephone.