My family has decided to take up Bunco as a “party game” and we played our first game this weekend. Great fun. But it got me to thinking of probability.

One of the suggested prizes on the prize sheet was $8 for whoever got an equal number of wins and losses. My dad and I thought “that will be a tough one to get” so we came up with a different way to distribute the cash if no one had an equal number of wins and losses. To our surprise, FOUR people got 12 wins and 12 losses.

To make is simple: the game has 24 total rounds. 12 people are playing. In each round you either get a W (win) or an L (loss). In each round, 6 people will get W’s and 6 will get L’s. What is the probability that you (a single player) will end up with 12 W’s and 12 L’s?

In our game, 4/12 people got 12 wins ans 12 losses. Does that mean their chances were 1/3?

Here’s an even trickier one, from the same game. I’m not sure if I can even describe the problem correctly:

You are randomly placed at 1 of 4 seats, at 1 of 3 tables (so essentially, 1 of 12 seats) at the beginning. This gives you a random partner. After each round, you and your partner go together to another table but you are no longer partners (you switch seats). So for example…

Round 1

A1&A2 - Wins (Table 1)

B1&B2 - Loses (Table 1)

C1&C2 - Wins (Table 2)

D1&D2 - Loses (Table 2)

E1&E2 - Wins (Table 3)

F1&F2 - Loses (Table 3)

Round 2

A1&C1 (Table 1)

A2&C2 (Table 1)

D1&E1 (Table 2)

D2&E2 (Table 2)

B1&F1 (Table 3)

B2&F2 (Table 3)

You keep mixing it up from there. Wether you win or lose determines which table you go to next, and you never keep the same partner 2 times in a row. Also in the above example, in Round 2 A1 could have ended up with C1 or C2, A2 with C1 or C2, D1 with E1 or E2, etc. It’s sort of random which person you end up with in the next round.

Is there some sort of probability formula that could apply to this situation to see if there is a probability of NEVER ending up with a specific person as a partner? I recall that in our game Saturday I only ended up with my best friend once, in the very last round. I don’t think I ended up with my mom at all.

I really suck at probability so I am not even sure if this is a real type of situation that can be written out in a forumula. But, it seemed interesting to me.