Probability of receiving a fixed bit sequence in a binary symmetric channel
I believe the answer can be derived as follows:

The probability of receiving 1111 is equal to the probability of transmitting a 1 taken to the fourth power.  The probability of transmitting a 1 is equal to the sum of the probability of transmitting a 1 given the input is a 0 plus the probability of transmitting a 1 given the input is a 1. Based on your information that is equal to:

\$0.4 \cdot 0.1 + 0.6 \cdot 0.9 = 0.04 + 0.54 = 0.58\$

Thus the probability of receiving 1111 is \$0.58^4 = 0.1132\$