A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2

# A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2

H
Asked by 2 years ago
240 points

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?

A communications
hansel

Answered by 2 years ago
8.7k points

What indepedence assumptions are you making

Assume that each digit received is independent of each other

what is the probability that the message will be wrong when decoded?

The message will be wrongly decoded the majority of binary digits form a majority of wrong numbers. So for instance, if we wanted to transmit a 1, but the person received 00011, then this would be wrongly decoded to 0. So if we let,

X = number of wrong digits

$$P(X \ge 3)=P(X=3)+P(X=4)+P(X=5)$$

is the desired probability which is a binomial probability,

$$=C(5,3)(0.2)^3(0.8)^2+C(5,4)(0.2)^4(0.8)^1+C(5,5)(0.2)^5(0.8)^0$$

$$\approx 0.00579$$

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$

Use LaTeX to type formulas and markdown to format text. See example.