1. 2 board, 18 stay 

(0.1) (1-0.2)^18 

2.  The 1 in the last transmat line.

3.  The total payouts are 4,8,12,13,14 if 1,2,3,4,5+ customer come in.
The 5 corresponding probabilities are dpois(i,lambda), i = 1,2,3,4 and
1 - ppois(4,lambda).

c(4,8,12,13,14) %*% c(dpois(1:4,lambda),1-ppois(4,lambda))

4.  

1:5 %*% inventory(0.6,0.4,5)