Probability Question with Random Variables perhaps.

[tennesseewiz]

Homework Statement

A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability 0.2. 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?

Homework Equations

This problem is in the Random Variables chapter of our book, but I don't see how we should use a random variable here. Maybe let X={the number of incorrectly received digits in a string of five equal digits}?

The Attempt at a Solution

I've been told the solution is 0.942, but I have no idea why. When I looked at it, what I wanted to do was:

(0.2)^5

Obviously that's wrong.

 

[HallsofIvy]

A message will be decoded incorrectly if and only if a majority of the bits get changed. What you have is the probability that all get changed.

However, 0.942 is NOT "the probability that the message will be wrong when decoded"- that should be obvious. If you only transmit one bit, the probability it is wrong is 0.2. Transmitting it five times won't make it more likely to be wrong! Perhaps it is the probability that it is decoded correctly.

What is the probability that 3, 4, or 5 bits get transmitted incorrectly? This is "binomial distribution" problem.

 

[Dick]

Yes. The number of incorrectly received digits. The message will come through correctly if there are 0, 1 or 2 incorrectly received digits. Use the binomial distribution.

 

[Focus]

I doubt the solution 0.942 is correct. For one the question says we reduce the error, from 0.2 to 0.942 is not really a reduction.

As for the solution, (0.2)^5 is not correct. Think about the different possibilities, if I was sending a 0 to you, and you received 00111 or 10111 what would you think I was sending? You don't need to transmit all of them wrong, just some. So sum up all the possible ways the transition can be wrong.

Use the random variable X={the number of incorrect digits in the string}. It should be clear what value X needs to be above for there to be an error.

Good luck.

 

[tennesseewiz]

Wait... isn't the problem saying that they are sending 5 0's in hopes that at least one 0 gets received? So, as long as you get one 0, it's like saying "what's the probability that you don't get any 0's?". I'm not understanding why I need to look at the probability that 3, 4, or 5 bits get transmitted incorrectly. Where are those numbers coming from?

Does this question require any kind of computer coding knowledge? Because I'm really confused about what "majority" coding is.

 

[Dick]

No. The problem is saying if you send 5 zeros, then what the probability that the majority of the received bits will be zero.

 

[tennesseewiz]

Oooohhh! Okay, let me go work on that, get an answer and see what you guys think.

 

[tennesseewiz]

So, I didn't get .942, but I got 1-.942 instead.

 

[tennesseewiz]

Oh, and thanks for all the help! I hate homework problems like these where I don't know what the questions is asking... :-s

 

[Dick]

So, I didn't get .942, but I got 1-.942 instead.

Right. 0.942 is the probability is was received correctly, not incorrectly.