# Probability message is sent correctly

A message of $$10^k$$ binary digits is sent along a fibre optic cable with high probabilities $$p_0$$ and $$p_1$$ that digits 0 and 1, respectively, are recieved correctly. If the probability of a digit in the original message being 1 is $$a$$, find the probability $$Pr$$ the entire message is received correctly.

I am stuck between two possibilities for this answer.

Approach 1: There are $$a10^k$$ 1's and $$(1-a)10^k$$ 0's. So $$Pr=P(\text{All 1's send correctly})P(\text{All 0's send correctly})=p_1 ^{a 10^k} * p_0 ^{(1-a) 10^k}$$

Approach 2: For a particular bit $$P(correct) = P(\text{It is a 1})p_1 + P(\text{It is a 0})p_2=ap_1 + (1-a)p_2$$ So, $$Pr = (ap_1 + (1-a)p_2)^{10^k}$$

I would be grateful for an explanation why one of these approaches is incorrect (or both, but I'm confident one of them is), as they both sound like they should work.

• The two methods are addressing different questions. In method $1$ we are assuming that exactly $a\%$ of the total were $1's$. In method $2$ we are assuming that each digits is $1$ with probability $a$, independent of all other choices. These are different questions.
– lulu
Commented Jan 6, 2021 at 18:06
• Does every bit have to be correct in order for the message to be received correctly? This isn't a fair assumption because such data is transmitted using error correction such as the Golay code. Commented Jan 6, 2021 at 18:12

To stress my earlier comment: this is a problem of semantics, not math.

The problem, as stated, admits two interpretations.

Interpretation $$I$$: The message contains exactly $$a\times 10^k\,$$ $$1's$$. In that case, your first approach is correct. Well, assuming that the noise hits each digit independently (but I think that's clearly the intent of the question).

Interpretation $$II$$: Each digit independently has a probability $$a$$ of being a $$1$$. In that case, the probability of getting a single digit right is $$ap_1+(1-a)p_0$$, just as you suggest in your method $$2$$.

I think either interpretation makes sense, so it's just a question of which one was intended. Your first method addresses the first interpretation, and the second method addresses the second.

• Thanks, this helps a lot. Could you please add to your answer what could be added to the quoted question to make it unambiguous(for both interpretations)? Commented Jan 7, 2021 at 9:25
• @janes If I intended the first, I'd say explicitly that exactly $a\%$ of the message consisted of $1's$. If I intended the second, I'd say that each digit was a $1$ with probability $a$, independent of all the other digits.
– lulu
Commented Jan 7, 2021 at 12:22