The Port Authority of a large East Coast City is investigating the traffic flow in and out of a large train station in the middle of the city. There are six entry gates and six exit gates that travelers can use to enter or leave the train station. An experiment is to be conducted to observe the number of gates open (i.e., a person is traveling through it) at a given point in time. You can assume that each point in the sample space is equally likely to occur, where an outcome is characterized by the two-tuple \((x, y)\) denoting that \(x\) entry gates are open and yexit gates are open.

a. Define an appropriate sample space, \(S\), for this experiment.

b. Define the appropriate probability set function for assigning probability to any event \(A \subset S\).

c. What is the probability that one entry and one exit gate will be open?

d. What is the probability that at least half the gates will be open in each direction?

e. What is the probability that the same number of gates will be open in each direction?

f. What is the probability that the total number of gates open will be less than four?