## Suppose that it takes 15 seconds to cross the street, and the
## probability that no car passes in any five second interval is a
## given probability p.  Calculate the average time it takes to cross
## the street, and the probability that it takes longer than one
## minute to cross.


## The probability that no car passes in any five second interval.
p = 0.7

## The number of simulation replications.
nrep = 1e3

## The total of all waiting times (including the 15 seconds required
## to cross).  
WT = 0

## The number of occasions on which more than 60 seconds was required to
## cross (including the 15 seconds that it takes to cross).
T60 = 0

## Simulation replications.
for (k in 1:nrep) {

    ## Indicates when cars are present for the first three time blocks.
    C = 1*(runif(3) < p)

    ## Determine how long it takes to cross in this simulation.
    n = 3
    while (TRUE) {

        ## Check whether we can cross.
        if (all(C)) { break }

        ## Update the status of the cars.
        C = c(C[2], C[3], 1*(runif(1)<p))

        n = n+1
    }

    ## Update according to what happened on the current simulation.
    WT = WT + n
    T60 = T60 + 1*(n>12) 
}


## Convert to a mean and a probability.
WT = WT/nrep
T60 = T60/nrep