A long queue in front of a Moscow market in the Stalin era sees the butcher whisper to the first in line. He tells her "Yes, there is steak today." She tells the one behind her and so on down the line. However, Moscow housewives are not reliable transmitters. If one is told "yes", there is only an \(80 \%\) chance she'll report "yes" to the person behind her. On the other hand, being optimistic, if one hears "no", she will report "yes" \(40 \%\) of the time.

If the queue is very long, what fraction of them will hear "there is no steak"? [This problem can be solved without finding a formula for \(P^{n}\) (here \(P\) is the transition matrix) - although you might find it a challenge to find the formula].