My husband works for the Federal government. He takes a van pool to and from his work on most days (he has a 120 mile commute round trip). The van pools are a great deal. If someone rides in a van pool, they commit to riding a certain number of days per quarter. They pay a membership fee and are reimbursed for saving gas.

In response to fraud, two significant changes were made to the van pool contracts:

- Van riders have to ride the van at least 50% of the working days
*each month*(instead of each quarter). - The van needs to be more than 50% full (not including the driver) at least 80% of the time.

**First requirement**

The first requirement is a problem. Vacations, has off-site training, goes to a conference, etc, interfere with ridership. Add two of these events together, and you end up with a severely inflexible policy. Employees who have to travel or who want to take a vacation may not be able to abide with the new requirements. Employees that travel a lot may not be the best candidates for the van pool, but employees who have one long trip *ever* should be able to ride the van.

**Second requirement**

The second change is even more restrictive. My husband’s van holds 14 people. They need to have at least 8 people ride (7 out of 13 + 1 driver) at least 4 out of 5 days in the week. It’s worth noting that small vans are going to have more of a problem with this because they typically have more variability in who shows up.

Looking at the long-term, a van needs to have at least 8 riders 80% of the time. That implies that the average number of days until the van is not “half” full needs to be 5. If we assume that each day is independent of the others, then we can model this as a geometric distribution. To get the desired average of 5, we need a “success” probability of 1/5. That is, the probability of a van being less than half full on any single day needs to be 20%. Is that realistic?

Let’s look at the 14 riders. Let’s say they each ride each day with probability p and that each day is independent. Then we can model the number of riders each day R~ Binomial(14,p). We need P(R<8) < 0.2 to meet requirement #2 (from the last paragraph). But if p=0.5 (as required), then P(R<8) = 0.605. We need p > 0.642 to get P(R<8) < 0.2.

Thus far, I’ve assumed that each day is independent. Making that assumption will yield optimistic results. The “optimistic” results suggest that even under idealistic assumptions, the new van pool requirements will be extremely difficult to meet.

But this assumption is not valid. Riders ride less on Mondays and Fridays (40% of the work week). In practice, this means a van is likely to have at least 8 riders on three days of the week (Tue-Thu), but not four as required. For simplicity, let’s say that a van always has 8 riders on Tue-Thu (an idealistic assumption, but not too unreasonable). Let’s say that each rider takes the van with probability p = 1/3 on Mondays and Fridays (a modest change), and let’s assume that each rider and day are independent. Then the van has 8 riders with probability 0.0576 on these days. *The probability that the van covers at least one Monday or Friday per week is 0.111*. *Therefore, under more realistic assumptions, it is even more unlikely that a van can meet assumption #2.*

Vans could meet these requirements if they “overbook” and allow more riders than there are seats in the van. They will surely need to risk turning riders away to meet their requirements if more than 14 show up some day. This may happen when gas prices change, since ridership increases when gas prices shoot up. However, my husband has noted problems meeting requirement #2 in the past couple of months when gas has been ~$4 per gallon.

**Van pools and (un)intended consequences
**

Van pools are intended to improve the environment by encouraging car pooling. Van pools ultimately create an incentive for people to live very, very far from where they work, thus leading to more fuel usage. In the end, I do not think they save much (or any) fuel from being consumed. But this is open for debate.

On the other hand, van pools are great if your significant other wants a career. It has made our “two body problem” quite manageable. Accommodating one’s spouse’s career is rarely cited as a benefit to van pools. I am grateful that van pools exist so that both my husband and I can have careers. I hope it’s still possible for him to carpool with new rules.

**Final thoughts**

I applaud the Federal government for trying to crack down on fraud. But I encourage them to create rules that make it possible for riders to follow the letter of the law. And maybe someone who has taken a course on probability should look over the next set of rules and crunches some numbers.

Do you ride a van? Please correct anything I got wrong in the comments and add your two cents about how they can be improved.