Jacks or Better

I recently went to Las Vegas and became intrigued with the game Jacks or Better (JoB). This was partly due to the fact that my fiance and I were fairly successful playing the game: we won $13 overall! I was also interested in the game because there is the illusion that you have excellent odds of winning. A hand of Jacks or Better seems very easy to get.

For those who don't know JoB, the rules are pretty simple: any hand with a pair of Jacks or better is a winner. The game can cost any amount to play, but here I focus on the $1 game. Winning hands with a $1 bet depend on the type of Jacks or Better, but they typically follow this pattern:

Jacks or better: $1
2 pair: $2
3 of a kind: $3
Straight: $4
Flush: $6
Full house: $9
Four of a kind: $25
Straight flush: $50
Royal flush: $250

There are many strategies already well documented online. I found the strategy below (I lost the link to the site with this strategy, but here's a link to a similar one). Basically, you want to hold cards in the order of rules listed.

Hold any winning hand of four cards or better
Hold any 4 cards to a royal flush (10, J, Q, K, A)
Hold any other winning hand
Hold any 4 cards to Straight Flush
Hold any 3 cards to a royal flush
Hold any 4 cards to a flush
Hold 2 of a kind
Hold any cards to an open straight
Hold any 2 high cards of the same suit
Hold any 3 cards to a straight flush
Hold a J, Q, and K of different suits
Hold any two high cards of different suits
Hold J, Q or K with a Ten of the same suit
Hold any single high card

With this strategy in mind, I was curious to see what the winnings would be over time. I wrote a program in python to simulate a person playing the game of JoB using the strategy listed above. I started the agent with an initial amount of $50. The agent played until he was below $50 or 50 times (which amounts to 50 plays). Over the course of play, I tracked how much the agent won or loss, and repeated this process 10,000 times. After 10,000 rounds, I plotted the results using matplotlib. The results are below:

As you can see, most of the runs fall between a gain of $50 and a loss of $50. There are much steeper gains than losses. Here are some statistics I gathered during the run:

Total Winnings: -$1.46
Number of Runs Zero or Above: 3734
Highest amount earned during a run: $258
Number of Runs Below Zero: 5973
Highest amount lost during a run: $32

What's so interesting about this is that the total winnings is practically zero! After 10,000 times of playing, the agent loses less than $2. I was curious to see if this small loss would hold over time, so, to the dismay of my computer, I ran the code a couple more times. The next two times, the agent lost $1.85 and $1.36. With such small gains for the casino, I'm wondering how much a casino makes off this game. I suppose that a good number of players do not know the best strategy, so this might tip there earnings even higher.

Also interesting about these results is the 5:3 ratio of losing to not losing. This would suggest that the agent should lose more. However, the agent can win a lot more than it loses. Each play only costs a dollar, but a win could be up to $250. Even with these large earnings, it's not enough to overcome the losing trend. It might seem like you have a good opportunity to win with Jacks or Better. However, in the long run, you will still lose like any other game in Las Vegas. :)

I've posted my code here for anyone to review. Also, please feel free to offer any suggestions for enhancements to the strategy or the code!