From Statistical Inference (2nd edition) by Casella Berger
The flow of traffic at certain street corner can sometimes be modelled as a sequence of Bernoulli trails by assuming that the probability of a car passing during any given second is a constant $p$ and that there is no interaction between the passing of cars at different seconds. If we treat seconds as indivisible time units (trials), the Bernoulli model applies. Suppose a pedestrian can cross the street only if no car is to pass during the next $3$ seconds. Find the probability that the pedestrian has to wait for exactly $4$ seconds before starting to cross.
The suggested answer is:
$$(1-p(1-p)^3)(1-p)^3$$
And the reasoning is that: the last three seconds must have no car passing (explains the last $(1-p)^3$). In considering what happened to the first four seconds, we have to exclude the situation in which the first-second having car passing and the last three seconds having no car passing (explains the term $1-(1-p)^3$). Assuming independence between two parts, we multiply these two terms together to get the probability required.
However, I find answer $(1-(1-p)^3)p(1-p)^3$ makes better sense. The reasoning is that if we want the pedestrian to wait for exactly $4$ seconds, the fourth second must have car passing, otherwise the pedestrian would cross the street one second earlier.
Could someone please explain to me why the first answer is the correct one and what are the points that I have missed when I provides the alternative explanation?
