*Four co-workers carpool to work each day. A driver is selected randomly for the drive to work and again randomly for the drive home. Each of the drivers has a lead foot, and each has a chance of being ticketed for speeding. Driver A has a 10 percent chance of getting a ticket each time he drives, Driver B a 15 percent chance, Driver C a 20 percent chance, and Driver D a 25 percent chance. The state will immediately revoke the license of a driver after his or her third ticket, and a driver will stop driving in the carpool once his license is revoked. Since there is only one police officer on the carpool route, a maximum of one ticket will be issued per morning and a max of one per evening.*

*Assuming that all four drivers start with no tickets, how many days can we expect the carpool to last until all the drivers have lost their licenses?*

Just a simulation from me this week. I’m sure there’s a nice analytical approach.

```
sim <- function(){
drivers <- 1:4
tickets <- replicate(4, 0)
n <- 0
while(length(drivers) > 0){
if (length(drivers) == 1){
driver = drivers
}
else{
driver <- sample(drivers, 1)
}
x <- runif(1)
if (x < (0.1 + (driver - 1)*0.05) ) {
tickets[driver] = tickets[driver] + 1
}
if (tickets[driver] == 3) {
drivers <- drivers[drivers != driver]
}
n = n + 1
}
return(n)
}
v <- c()
for (i in 1:10000){
v <- c(v, sim())
}
mean(v)
```

`## [1] 76.9298`

So the carpool should end on the 38th day.

And here’s a histogram.

```
library(ggplot2)
library(ggthemes)
df <- as.data.frame(v)
ggplot(df, aes(v)) +
geom_histogram(bins=40) +
theme_fivethirtyeight()
```