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I am looking for a way to calculate the high or low odds for each card turn in a game of "Play your cards right". You can play this game and see what I mean here

Basically, there are several rows of 3 cards face down, the bottom row has 4 cards. The first card of the bottom row is turned over, and you have to guess if the next card will be higher or lower. If you guess correctly, you then guess if the next card will be higher or lower and so on. When you reach the last card in that row, it is moved to the first spot of the next row up, and you go through the same guessing process again.

enter image description here

As you can see from the image, the higher and lower buttons have the amount you can win for each guess.

They are not always the same for the same card, for example "Jack" in the image above has you winning 3.30 for higher, but if I was to encounter the Jack further in the game, like the 5th card, the amount for higher would be different.

The rest of the rules are Aces are high, and a draw loses

I would like to know how these odds are calculated for a game of 16 cards. Also does anyone know what the house edge is on a game like this?

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Assuming that the cards are drawn at random from a single standard 52 card pack:

There are 51 unseen cards, and the next card could be any of them with equal probability. There are thirty-six cards lower than a jack, twelve higher, and three equal. Thus, $$P(lower) = 36/51\\P(higher)=12/51\\P(same)=3/51.$$

The returns you would want on these bets for them to be an even-money proposition would be the reciprocal of these; 51/36 and 51/12 respectively. So they are only paying 1.14 for "lower" when 51/36 (about 1.42) is the "fair" rate and only paying 3.30 for "higher" when 51/12 (=4.25) is the fair rate. This is a fairly significant house edge, over 20%.

This changes as more cards appear, because there are different numbers of higher and lower cards left. For example, suppose a game starts with the King, Queen and Jack of hearts. Now there are 49 cards left instead of 51. The chances of (higher, lower, same) have become (10/49, 36/49, 3/49) so the given odds change to reflect that.

If you got to see the specific 16 cards in play at the start, then that changes things.

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  • $\begingroup$ Excellent, just what I was looking for! $\endgroup$ – superphonic Oct 17 '16 at 10:44
  • $\begingroup$ Just to add, In hilo usually Ace is low, so there would be 40 cards lower than a Jack. Great answer, very useful. $\endgroup$ – Gary Aug 27 '17 at 17:56

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