# The Mathematics of a Lowball Sidebet

This month I have gone ahead and gone along with the other Lowball players wishes and played the “gidgets and gadgets” of Lowball sidebets. I figure it doesn’t cost me anything, it’s good for my image, and it pays to be liked by the other players which will tend to happen when they go along with their stupid bets.

A younger player has learned the game and proposed a new side bet. He was open to taking either side of the bet. The bet is that one player will pay another play \$20 if he doesn’t have a King. However, if that player does have at least one King, then he will be paid \$35 per King by the other player.

Let’s take a look at the math on that. Since we’re dealing with a 53 card deck using the Joker, the total number of five card hand combinations are 53 choose 5 which total 2,869,685. The total number of hands which involve no Kings are 49 choose 5 which equals 1,906,884. So the odds of getting a hand without a single King are 66.4%. We can alos tell that there are exactly 2,869,685 – 1,906,884 = 962,801 hands containing a King.

The total number of hands which involve getting one King are 4 choose 1 times 49 choose 4 which is: 4 x 211876 = 847504.
Divided by the original 2,869,685 we see that the chances of getting a hand containing a single King are 29.5%.

For a hand with two Kings, it is 4 choose 2 multiplied by 49 choose 3: 6 x 18424= 110,554
For a hand with three Kings, it is 4 choose 3 multiplied by 49 choose 2: 4 x 1176 = 4704
For all four Kings, there is only one combination for the four multiplied by 49 = 49

So if we add up all of the the hands with Kings, we get 962,811 hands containing a King: ten more than we were expecting. I’m guessing there we some rounding errors in dealing with these large numbers. When we’re accounting for close to 3 million hands, having a 10 hand discrepancy ain’t too bad. So I’m just going to convert to percentages.

66.4% of hands contain no King. 29.5% contain 1 King. 3.8% of hands contain two Kings. .16% contain three Kings, and .002% containing 4 Kings.

Going back to the original wager, if we play out 10,000 hands, then we’d have:
10,000 x 66.4% = 6640 No Kings receiving \$20 = \$132,800
10,000 x 29.5% = 2950 with one King receiving \$35 = -\$103250
10000 x 3.8% = 380 with two Kings receiving \$70 = -\$26,000
10000 x .16% = 16 with three Kings receiving \$105= -\$1680
And quad Kings being such a rare event that it doesn’t figure into 10,000 hands.
So if we do the math, over 10000 hands, the person buying the Kings will reap a profit of
\$1870 which amounts to 19 cents a hand or so.

Nothing to get excited about, but the kind of stuff that gets the chips flying.