Scratching your post-pandemic itch to travel, you decide to accept an internship offer at the Paris headquarters of BlaBlaCar (https://en.wikipedia.org/wiki/BlaBlaCar Links to an external site.andhttps://www.blablacar.com/)Links to an external site.,an online marketplace for carpooling with 70 million active users. BlaBlaCar matches people that want to go from (near) A to (near) B at a given time (for example, from The Hague to Rotterdam between 7 am and 8 am on some Friday) with drivers that are driving from A to B around that day and time.

Unlike other ridesharing platforms, most BlaBlaCar drivers use the platform to cover travel expenses and not as a source of income. Drivers post their ride trajectories some time in advance, and interested riders contact the drivers to arrange pickup details. BlaBlaCar algorithms suggest prices to drivers (usually enough to cover travel costs), and BlaBlaCar makes money by charging drivers 15% of the fare.

One of your first tasks is to estimate the number of drivers and passengers needed to make BlaBlaCar's market "thick." To do so, you devise a simple stylized model of this platform to gain some intuition. To simplify things, in this "first cut" model, you make a few assumptions:

1. A day is divided into 24 slots, each one hour in length;
2. You consider only one route on one specific day (for example, The Hague to Rotterdam and the 24 1-hour slots on February 27th);
3. You assume that, at some point before the specific day you are analyzing, drivers and riders arrive simultaneously on the platform. Furthermore, each rider and driver choose only one slot on that day.
4. The platform matching mechanics are as follows: (a) riders and drivers enter the platform and choose specific slots. (b) In each slot, drivers and riders are matched at random. (c) unmatched rivers and riders leave the platform. For example, if there are 5 riders and 7 drivers in a given slot, there will be 5 matches, and 2 drivers will be unmatched and will leave the platform. Similarly, if there are 1 rider and 0 drivers, the rider will not be matched (and will not change time slots) and leave.
5. Each driver can only take one passenger. For example, if there are two drivers in one slot and only one passenger, or two passengers in a slot and only one driver, only one ride occurs.
6. Drivers and passengers choose slots with the same probability - all slots in a day are equally likely to be selected (they are selected uniformly at random).

Question 1 (10 points)

If a single passenger arrives on the platform and is looking for a ride on a specific slot and route on a given day, what is the minimum number of drivers that should be in the system offering rides for that route so that they find a ride with a probability of at least 80%?

Question 2 (40 points)

Note: This question is more advanced, and I suggest using Excel, R, Python, Matlab (or your preferred programming language) to solve it.

Note that the scenario in Question 1 is great for the passenger but terrible for the drivers. A well-managed platform with many matches depends not only on the number of drivers and passengers but also on the balance between them. A liquid platform is one where many matches occur, and a small number of people are unmatched. This concept is also known as market thickness. Thus, with the same assumptions as before, what is the minimum number of drivers and passengers that must arrive at the platform searching for a trip on a day and route such that, in expectation, at least 80% of passengers and 80% of drivers find a match?

Question 3 (5 points)

Of course, this model is somewhat unrealistic. However, do you think it provides an upper bound or a lower bound on the actual number of drivers needed?

Question 4 (5 points)

How could you make this model more realistic? What other data would you need?