>  p(mud | rain) = 1.0
  >  p(rain | mud) = p(rain)

I'm not sure these statements say what you think they do. Assuming P(rain) > 0, the first statement

  p(mud | rain) = 1.0

is equivalent to

  p(mud & rain) = p(rain).

The second statement therefore implies

  p(rain | mud) = p(mud & rain).

Thus

  p(rain & mud) / p(mud) = p(mud & rain).

This, however, leads to the surprising conclusion

  p(mud) = 1.0.

According to your theory, it is always muddy.