Bus Driver drinking game analysis

Quick pause to teach you all a new drinking game, and have some statistical analysis of it.

The game I am about to describe and analyze is actually the second part of the whole game. It is the penalty part. The person who plays it is the one who lost the previous part, and now has to face the consequences. That person is the bus driver.

To simplify, I will skip the first part, since it could be basically any other game or dispute.

The idea of the bus driver game is: 6 cards are initially dealt to the bus driver. For every J, Q, K, A on these cards, the driver has to drink the number of shots equivalent to that card, which is:

• 1 for J
• 2 for Q
• 3 for K
• 4 for A

Additionally, the number of shots is also converted into new cards to be drawn, and follow the same rules. So, if a Q is drawn, 2 more cards need to be dealt. The bus trip will go on until no J, Q, K or A are drawn, which is when there are no more passengers, and the driver finally reaches its destination.

Assuming a normal deck of cards, there are 16 problematic cards (4 of each J, Q, K and A), and the maximum number of shots a person could suffer would be the sum product of each of the 4 cards by their assigned value, so (4*1+4*2+4*3+4*4), which give us 40 shots.

So, a few questions for me to answer:

• How many shots is a person expected to take?
• How likely is for a person to drink no shots (lucky)?
• How likely is for a person to drink 40 shots (unlucky)?

I made a script, for which the code I will share below, to run a few Montecarlo simulations. Feel free to open the link and study them if it is something you are interested into. The Montecarlo code was done in C#, and the analysis was done in Python.

A pretty ugly (and honestly unexpected) distribution for the probability of each number of shots, with the probability of 0 shots being 9.6% and 40 shots being 1.26%. The expected number of shots the poor driver has to take came out at about 13.4 shots.

Now, the difficult question... How difficult would it be for it to be solved analytically?

Below are the code sources of the quick study, any comment and insights are welcome.

https://github.com/ramoyses/busdriver