To cross a single lane of moving traffic, we require at least a duration d. Successive car interarrival times are independently and identically distributed with probability density function fT (t). If an interval between successive cars is longer than d, we say that the interval represents a single opportunity to cross the lane. Assume that car lengths are small relative to intercar spacing and that our experiment begins the instant after the zeroth car goes by. Determine, in as simple form as possible, expressions for the probability that:
(a) We can cross for the first time just before the nth car goes by.
(b) We shall have had exactly m opportunities by the instant the nth car goes by.
(c) The occurrence of the mth opportunity is immediately followed by the arrival of the nth car.