Some useful mathematics

Grant and I had an argument about some probability when we were playing a drinking game a while back. I said I was right, and I said I would prove it. So here I am, proving it. ;)

In the game of "up the river, down the river" each player is dealt four cards face up one at a time. For the first card dealt, each player tries to guess if it will be red or black (suit). Guessing right means you win. For the second card, each player is asked to pick higher or lower (in rank compared to their first card dealt). For the third card, you have to guess inside (between the two first cards) or outside (higher or lower than both). For these second and third questions, a tie means you loose. And of course it's a drinking game so every time you loose you drink. Then for the fourth card, you guess which suit.

Then eight cards are dealt face down, and you turn them up one at a time, in one column, you have to drink for each card you have that matches, and in another column, you get to give drinks to the person(s) of your choice. What's more, the number of times you have/get to drink/give is 1, 2, 3, 4 depending on the row. (Sometimes this is 2, 4, 6, 8, depending on the crowd).

Note: with this game the number of people, call it p, must satisfy 4p + 8 <= 52. In other words, you can play with at most 11 people. (Incidentally, this is the same number of cards per person and community cards, counting the three burn cards, as in the game Omaha.)

When I'm dealing, if there's enough cards, because there are few enough people playing, I like to deal a fifth card, asking even or odd. Either everyone drinks or no one drinks on an ace because it is both 1 and 14. This can only be done when 5p + 8 <= 52, or when there are eight or less people. (Unfortunately, I don't know a poker game where you get five hole cards and five community cards.)

Grant doesn't like the fifth card though. He thinks (or at least he used to) that it makes it less likely that on of your four matches the eight, because some of your "outs" (term from poker meaning a card that would make your hand, in this case, any card the same rank as one of your four) will be dealt when the fifth card is dealt around.

It is perfectly understandable that one might feel that they are less likey to pair one of their cards to the board when everyone gets dealt a fifth card. It is more likely that you will see all four cards of one rank, and then you know you have no chance of pairing that card. And if two of those four are in your hand, you might really be mad, because that fifth card that someone got, if it is the last card dealt to eight players, could have been a give eight, and you would've had two of those, so you would've gotten to dish out 16 drinks!

Now I'd like to show that your odds are greater to give and take more drinks with five cards.

Proof
Consider the game above, dealt with four cards to each player. Now, suppose that after you play the game (with eight or less people) when all of the river cards are face up and all the drinks have been dealt, you deal everyone an extra card, and if it matches one of the river cards, you get to give/take the corresponding number of drinks. In some cases drinks will be given or taken. For example, if you get dealt a seven of hearts, and the seven of diamonds was the "give 2" card. You would get to give 2. Now more drinks have been given. And the only thing we did was deal a fifth card. This is not even considering that some people will guess the "even or odd" part correctly. So clearly, in the game "up the river, down the river," more drinks are given when you play with "even or odd" (the fifth card). ■

Stay tunned for another drinking game related discussion on the topic of "racing aces."