Solitaire Odds
September 28th, 2005 by SplineGuy
I have finally taken the time to answer a question that has bothered me for some time. There is a solitaire card game that I learned while I was a Sojourner (high school summer missionary) in Bakersfield, California. I don’t know any sort of official name for the game, which means I can’t refer you to any site that describes it. I’ve called it either One in a Million or Bozo (a name provided by a Jennifer (Taylor) Mykitiuk while she was in Junior High.)
There is no strategy at all and so, winning is pure chance. I wanted to know what chance. Here is the approach to the game.
- After thoroughly shuffling, hold the deck face down.
- One at a time draw four cards from the bottom of the deck and place them on the top of the deck each face up.
- Looking at the fourth card and the first card, if they are the same suit, discard the middle two cards. If they are the same denomination, discard all four. Otherwise, move on to the next step
- Draw the next two cards (or four if all have been discarded) from the bottom of the deck, placing them face up on the top. Comparing the top card and the fourth from the top, repeat step 3.
- Continue drawing and discarding until all cards have been drawn.
The goal is to discard all the cards. In playing an awful lot since I learned the game in ‘94, I imagine I have won around 5 or 6 times total.
So, whis is the actually likelihood of winning. My first attempt of deriving an exact probability distribution for the outcomes failed miserably. Thus, I moved on to a “Monte Carlo” simulation approach.
I designed a computer program that played the game and determined the remaining number of cards after randomizing a deck. Using the program, I can play the game up to 100,000 over the timespan of about 5 minutes. The following is the outcomes from 100,000 random simulations:
| Number Left | Frequency | Approx. Probability |
| 0 | 5,966 | 5.966 % |
| 2 | 20,261 | 20.261% |
| 4 | 22,494 | 22.494% |
| 6 | 10,988 | 10.988% |
| 8 | 12,201 | 12.201% |
| 10 | 5,921 | 5.921% |
| 12 | 6,903 | 6.903% |
| 14 | 3,295 | 3.295% |
| 16 | 3,732 | 3.732% |
| 18 | 1,836 | 1.836% |
| 20 | 1,967 | 1.967% |
| 22 | 945 | 0.945% |
| 24 | 1,086 | 1.086% |
| 26 | 518 | 0.518% |
| 28 | 590 | 0.590% |
| 30 | 285 | 0.285% |
| 32 | 300 | 0.300% |
| 34 | 172 | 0.172% |
| 36 | 170 | 0.170% |
| 38 | 84 | 0.084% |
| 40 | 93 | 0.093% |
| 42 | 42 | 0.042% |
| 44 | 56 | 0.056% |
| 46 | 21 | 0.021% |
| 48 | 18 | 0.018% |
| 50 | 16 | 0.016% |
| 52 | 40 | 0.04% |
From this, we can determine that the expected value (average outcome) is 7.3712 cards and a standard deviation of 6.57 cards. It seems that the real challenge of the game is not to get rid of all of your cards but to maximize the number of cards left behind. I have never had a hand where no cards could be discarded. That’s my new goal. You know, in my free time, that is, the time I am not spending programming monte carlo simulations.
