Question

A communications channel transmits the digits $0$and $1.$ However, due to static, the digit transmitted is incorrectly received with probability $2.$ Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit $00000$ instead of $0$ and 11111 instead of $1.$ If the receiver of the message uses “majority” decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?

Short answer

The probability that the message is wrong when decoded is$0.0579.$

Step-by-step solution

Given Information

The probability that the digit transmitted incorrectly is, $0.2.$

To reduce the chance of error $5$digits are transmitted instead of $1$digits.

The message is wrongly received when an$3,4$or$5$digits are transmitted incorrectly.

Solution of the Problem

The probability that the message will be wrong when decoded is,

$P(Wrongmessage)=P(X\ge 3)$

$=\left(\begin{array}{l}5\\ 3\end{array}\right)(0.2{)}^3(0.8{)}^2+\left(\begin{array}{l}5\\ 4\end{array}\right)(0.2{)}^4(0.8{)}^1+\left(\begin{array}{l}5\\ 5\end{array}\right)(0.2{)}^5(0.8{)}^0$

$=0.0512+0.0064+0.0003$

We get,

$=0.0579$.

Final Answer

The probability that the message is wrong when decoded is$0.0579.$