Question

A message source M of a digital communication system outputs a word of length 8 characters, with the characters drawn from the ternary alphabet {0,1,2}, and all such words are equally probable. What is the probability that M produces a word that looks like a byte (i.e., no appearance of ‘2’)?

Answer 1

Answer:

The probability that M produces a word that looks like a byte is $(\frac{2}{3})^8$.

Step-by-step explanation:

It is given that a message source M of a digital communication system outputs a word of length 8 characters.

The characters drawn from the ternary alphabet {0,1,2}, and all such words are equally probable.

Total possible outcomes is

$3\times 3\times 3\times 3\times 3\times 3\times 3\times 3=3^8$

We need to find the probability that M produces a word that looks like a byte (i.e., no appearance of ‘2’). It means only 0 and 1 are included in the word.

Total favorable outcomes is

$2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=2^8$

The formula for probability is

$p=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$

$p=\frac{2^8}{3^8}$

$p=(\frac{2}{3})^8$

Therefore the probability that M produces a word that looks like a byte is $(\frac{2}{3})^8$.