I need some help with these two probability questions:
Cathy is trying to communicate information to George over a faulty network. The conveyed information is in the form of bits. The network is messy and so every ‘1’ that Cathy sends is received correctly with probability 0.8 (so with probability 0.2 any ‘1’ sent by Cathy is received as a ‘0’ by George). Similarly, every ‘0’ sent by Cathy is received correctly with probability 0.7 (so with probability 0.3 the ‘0’ sent by Cathy is received as a ‘1’ by George). Suppose that Cathy is thrice as likely to send a ‘1’ as she is to send a ’0’.
(a) Find the probability that George receives a ‘1’.
(b) If George receives a ‘0’, find the probability that Cathy intended to send a ‘0’.
For part a, I would assume you would use the law of total probability, since there are two different scenarios, and each have a specific probability for George receiving a 1. However, I'm not sure how to set this up.
For part b, I would assume you would use the Bayes Rule, since you need to find a probaility of A given B occurs. I originally formatted A to be the probability Cathy intended to send George a '0', while B is the proability that George actually receives a ) from Cathy. However, I also had trouble setting this problem up.
Sometimes, if using Bayes's Theorem defeats your intuition, it may help to imagine a perfectly representative sample of cases. Suppose that Cathy sends George $40$ bits. (Why not Alice and Bob? I don't know.) Then she tries to send $30$ ones and $10$ zeros. Of the $30$ ones, $24$ are received as ones and $6$ as zeros. Of the $10$ ones, $7$ are received as zeros and $3$ as ones.
So George receives $24+3 = 27$ ones, of which $24$ were intentional, and $3$ were intended as zeros. He also receives $6+7 = 13$ zeros, of which $7$ were intentional, and $6$ were intended as ones. The desired probability for (a) is therefore $27/40$, and for (b) is therefore $7/13$.
Can you convert each of these concrete workings into the corresponding expressions involving total probability and Bayes's Theorem?