Bits are sent through an information channel where each bit has $0.1$ probability of being transferred incorrectly, independent from other bits. (for example $0$ is sent and $1$ is received).
Bits will be sent in sequence consisting of $5$ bits. Bits from this sequence will be decoded according to the most frequent bit in sequence
Examples
Sent $\rightarrow$ Recieved
$00000 \rightarrow 0$
$01010 \rightarrow 0$
$01101 \rightarrow 1$
$11111 \rightarrow 1$
Questions:
(a) What is probability for recieved sequence being decoded correctly ?
(b) When $1$ million bits are sent over an information channel using sequences, what is expected value for incorrectly decoded sequences ?
Best Answer
Note that if $X$ is the number of bits that are transferred correctly, then $X$ follows a binomial distribution with parameters of selecting $5$ bits at a time and $0.9$ probability of a bit having been transferred correctly:
$$ \operatorname{P}(X=r)=\binom{5}{r}(0.9)^r(0.1)^{5-r} $$
For a sequence to be decoded correctly, only three of the five bits need to be transferred accurately. Thus the probability that the sequence is decoded correctly is
$$ \operatorname{P}(X\ge3)= \operatorname{P}(X=3)+ \operatorname{P}(X=4)+ \operatorname{P}(X=5) =\sum_{r=3}^5 \binom{5}{r}(0.9)^r(0.1)^{5-r}$$
I’ll leave that arithmetic to you.
The probability that a sequence is decoded incorrectly is $1- \operatorname{P}(X\ge3)$, which we just solved. Expected value for number of successes follows the form (number of trials)×(probability of success). Thus you can expect
$$\frac{10^6}{5} \left[ 1- \operatorname{P}(X\ge3)\right]$$
sequences to be decoded incorrectly since there are $\left. 10^6 \middle/ 5 \right.$ sequences transmitted.