Q. 4.43 - A First Course in Probability

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? 

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: Solution of the Problem

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

$P( Wrong message )=P(X \ge 3)$

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

$=0.0512+0.0064+0.0003$

We get,

$=0.0579$.

Step 3: Final Answer

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