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 Aaron 2 years ago

**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,

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

$$\approx 0.00579$$

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Asked: 2 years ago