A few weeks ago we covered basic probability and house edge. In this article, we are going to look at a chain of chances with or without memory.

A basic example of this is a betting proposition of 4 coins in a row. The player has to call heads or tails and if all four land on that call, he is paid 10:1. So if the player bets 1 unit and wins, he gets back 10 units in winnings plus his original stake. This actually sounds a doddle, four coins with two sides each. How hard can it be to win? The chances multiply with each toss of the coin. 1st throw x2, 2nd throw x4, 3rd throw x8 and final throw x16. There are 16 different permutations of the sequence, it’s basic binary. Of these 16 permutations the player owns 1. So on the balance of probabilities, the house will pay out 11 units for every 16 units they take.  This gives the house an edge of 5 in every 16 sequences, which is 31.25%

Now lets look at a similiar bet, but with a pack of 52 playing cards. This time the bet is higher or lower than a 7, no cards returned to the deck, one player, aces low, pictures high. So out of each deck there are 24 cards out of 52 which fit the player’s call. Each time a successful card is drawn, the number of other winning cards in the deck goes down as well as the total number of cards in the deck. This is a game that has a memory, because a shrewd player could count the next higher or lower bet with increasing probability of winning a decent sum by the end of the deck. Let us make this a bit more interesting and offer odds of 50:1 if the players call applies to ALL 5 of the cards drawn from the fresh deck. We already know that any 7 drawn is an immediate loss as well as the other 24 cards from the opposite call. Each time a new card is drawn there are still 28 cards against the player and 1 less in the deck in their favour.

For 5 cards the favourable side of the deck would progress as:

24 / 52
23 / 51
22 / 50
21 /49
20 / 48

If you multiply all these numbers together you come up with just under 1 in 61 chance of winning. The win would pay out 50:1, netting the player 51. So on a balance, the house owns 10 out of every 61 wagered or 16.59%.

The game becomes a little more scarey to calculate if you took bets on each draw, side bets on the 7 and didn’t deal all the cards in each deck. Players could formulate a count system that worked out the swing of the deck and how much to bet, more in a later article, but do try it at home.