One-trick “Oh Hell” – what are you chances of winning?

On a holiday day Meaghan Fowlie and I decided to figure out your chances of winning in a one-trick “Oh Hell” game. “Oh Hell” is a variant to the Whist game; if you want to know the rules in more detail, check out the Wikipedia page.

So let’s assume that there is only one trick to play (each player has one card). Trumps are decided by a card that is flipped face up (so we know that is one less card in the trump suit in the game). What are the chances that you’ll win this trick? It depends of course on which of the following situations you are in:

– You don’t have a trump, but you are leading (you are the first to play). This is marked as nontrump.leading in the graph below.

– You don’t have a trump, and you are not leading. This is marked as nontrump.not.leading.

– You have a trump. Furthermore, it is higher than the card that is turned face up. This is called trump.turned.card.below.

– You have a trump, and it is lower than the card that is turned face up. This is called trump.turned.card.above.

If you’ve decided what case you are in, you can use the graph below to decide how likely you are to win the trick. The horizontal panels indicate the number of players (between 2 and 6). The x-axis indicates the card you hold. The lines indicate the probability that you’ll win. So if you play “optimally”, you’d bid 1 if the chance is greater than 0.5 and 0 otherwise.

 

ohell_have_trump

Of course, this is true only if you have no other information. Once other players place their bids, your estimation of your chances of winning should change. Maybe we’ll address this some other holiday day.